Ageing Concepts

Abstract

A considerable part of reliability theory is dedicated to the study of ageing concepts, their properties, implications and applications. In this chapter, we review some of the important results in this area and translate the basic definitions to make them amenable for a quantile-based analysis. Ageing represents the phenomenon by which the residual life of a unit is affected by its age in some probabilistic sense. It can be positive ageing, negative ageing or no ageing, according to whether the residual lifetime decreases, increases or remains the same as age advances. Generally, one investigates whether a given ageing concept preserves certain reliability operations such as formation of coherent structures, mixtures and convolutions. We first introduce the basic ideas behind convergence, mixtures, convolutions, shock models and equilibrium distributions. The ageing concepts are studied under three broad categories—based on hazard functions, residual life functions and survival functions. The IHR, IHR(2), IGHR, NBUHR, NBUHRA, SIHR, IHRA, DMTTF, IHRA* t ₀ classes and their duals along with their properties come under ageing notions related to the hazard function. In the class of concepts based on residual life, we discuss DMRL, DMRLHA, UBA, UBAE, HUBAE, DRMRL, DVRL, DVRLA, NDMRL, NDVRL, IPRL-α, DMERL classes and their duals. Those defined in terms of the survival function include NBU, NBU-t ₀, NBU* t ₀, NBU(2), SNBU, NBUE, NBU(2)-t ₀, NBUL, NBUP-α, NBUE, HNBUE, $\mathcal{L}$-class, $\mathcal{M}$-class and the renewal notions NBRU, RNBU, RNBUE, RNBRU and RNBRUE. A brief discussion is also made on classes of distributions possessing monotonic properties for reliability concepts in reversed time. The ageing properties of the quantile function models introduced in Chap. 3 are presented. Finally, some definitions and results on relative ageing are detailed.

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4.1 Introduction

A considerable part of reliability theory is dedicated to the study of ageing concepts, their properties, implications and applications. We review here some important results in this area and translate the basic results to make them amenable for a quantile-based analysis. We recall that the lifetime of a unit is the time up to which the unit performs as it is required to do. The age of the unit is the time up to which the unit fails to function. By the term ageing of a mechanical or biological unit with a lifetime distribution, we mean that the residual life of the unit is affected by its age in some probabilistic sense. This description includes the cases in which a unit experiences no ageing, positive ageing or negative ageing. Positive ageing simply means that the residual lifetime decreases when the age of the unit increases and it reflects an adverse effect of age on the random lifetime of the unit. This is a common feature of equipments or mechanical systems that are subject to gradual wear and tear while in operation. Negative ageing, which is the dual concept of positive ageing, has a beneficial effect on the life of the unit as age progresses. A well-known example is that of human beings where the average remaining life increases after infancy. This is also the case with mechanical devices that suffer from designing or manufacturing errors or operation by inexperienced hands initially which improves their performance later, as well as systems that undergo preventive maintenance which reduces wear out failures. No ageing applies to units whose remaining life continues to be the same at all ages. This situation is identified by equating the lifetime distribution with its residual lifetime distribution, i.e.,

$$\displaystyle{ \frac{\bar{F}(x + t)} {\bar{F}(t)} =\bar{ F}(x)\qquad \forall \ x,t. }$$

(4.1)

It is known that the exponential distribution is the only lifetime model that satisfies (4.1). Sometimes, (4.1) is referred to as the lack of memory property of the exponential distribution, and it plays a key role in the definitions of various ageing criteria.

There are several advantages in studying ageing concepts. Primarily, they reveal in some sense the pattern in which a unit deteriorates or improves in its functioning with respect to age. This, of course, imparts valuable information about the reliability of a unit. Secondly, life distributions are classified on the basis of their ageing behaviour. In practice, a knowledge of this ageing behaviour could be used effectively for model selection. Nonparametric inferential methods generally make few assumptions about the population from which observations are drawn. Ageing behaviour is an advantage in such situations, especially when the ageing class has geometric properties like convexity, star shapedness, etc. Moreover, these classes have several desirable properties in their own right such as bounds on the reliability function, distributional properties like unimodality, inequalities among moments and preservation properties under various reliability operations. Further, the identification of the no ageing property of the exponential law becomes a base in the construction of tests of hypothesis of exponentiality against different ageing criteria. Often, reliability classes, with appropriate interpretations, form the basis of model selection in other disciplines. While discussing different ageing classes and criteria, we distinguish them as based on the hazard quantile function, residual quantile function and some other properties. The quantile-based definitions of the ageing concepts discussed here are available in Nair and Vineshkumar [454].

4.2 Reliability Operations

Before defining ageing concepts, we provide some preliminary details which are vital to reliability theory. It is customary to investigate whether a given life distribution possessing a specific ageing property preserves it under various reliability operations. Important among the operations are formation of coherent structures of independent components, addition of lifelengths or convolution and formation of mixtures of distributions.

4.2.1 Coherent Systems

Coherent systems, introduced by Birnbaum et al. [103], helped to a great extent in laying the foundation of reliability theory as a separate discipline. Here, we consider systems that can be in only one of two states, functioning or failed. A structure function ϕ of a coherent system of n components (or a coherent system of order n) is such that, with x _i  = 1 if the ith component functions and x _i  = 0 if it fails,

• (i) $\phi (1,1,\ldots,1) = 1,$

• (ii) $\phi (0,0,\ldots,0) = 0,$

• (iii) $\phi (x_{1},x_{2},\ldots,x_{n}) \geq \phi (y_{1},y_{2},\ldots,y_{n})$, whenever $x_{i} \geq y_{i}$, $i = 1,2,\ldots,n$.

For example, the structure function of series and parallel systems described in Chap. 1 are

$$\displaystyle{\phi (x_{1},x_{2},\ldots x_{n}) = x_{1}x_{2}\ldots x_{n}}$$

and

$$\displaystyle{\phi (x_{1},x_{2},\ldots x_{n}) = 1 - (1 - x_{1})(1 - x_{2})\ldots (1 - x_{n}),}$$

respectively. The states of the components of a coherent system are represented by Bernoulli random variables and the reliability function of the system is then

$$\displaystyle{R(p_{1},p_{2},\ldots p_{n}) = E(\phi (X_{1},X_{2},\ldots,X_{n})),\quad \text{with}\ p_{i} = P(X_{i} = 1),}$$

where $X_{1},X_{2},\ldots,X_{n}$ are independent. Introducing a concept of time T now, let X(t) = 0 or 1 depending on whether the unit is failed or functioning at time t, we have

$$\displaystyle{ P(X(t) = 1) =\bar{ F}(t) }$$

(4.2)

as the survival function of T. For a coherent system of order n, $\phi (X_{1}(t),\ldots$, X _n (t)) is the performance process and the system life T has survival function

$$\displaystyle{\bar{F}(t) = R(\bar{F}_{1}(t),\ldots,\bar{F}_{n}(t)),}$$

where $\bar{F}_{i}$ is the survival function of the component lifelengths as defined in (4.2). For further details and properties, one may refer to Barlow and Proschan [69].

4.2.2 Convolution

Let X and Y be two independent random variables with distribution functions F(x) and G(x). The distribution of X + Y, specified by the distribution function

$$\displaystyle{ H_{1}(x) =\int _{ 0}^{x}F(x - t)dG(t), }$$

(4.3)

is referred to as the convolution of X and Y, and is usually denoted by H ₁ = F ∗ G. In the reliability context, convolution is interpreted as the operation of adding lifelength X to Y. When a spare part is available as the replacement of the original part, in case of failure of the latter, the two together acts as a system of two components and the available total lifetime in this case is X + Y, the sum of the lengths of lives of the original and spare parts. Also, if the component with life Y fails at any time t preceding x, while component with life X fails during the remaining interval of time x − t, then (4.3) is the probability that the sum X + Y does not exceed x.

4.2.3 Mixture

Assuming that F(x | θ), θ ∈ Θ, be a family of distributions and G(θ) be a distribution on Θ, the mixture of F with G is given by

$$\displaystyle{ H_{2}(x) =\int _{\varTheta }F(x\vert \theta )\,dG(\theta ). }$$

(4.4)

The hazard rate of (4.4) is

$$\displaystyle{{h}^{{\ast}}(x) = \frac{\int _{\varTheta }f(x\vert \theta )dG(\theta )} {\int _{\varTheta }\bar{F}(x\vert \theta )dG(\theta )}.}$$

If the hazard rate of F(x | θ) is bounded by $h_{1} < h(x\vert \theta ) < h_{2}$ for all θ in the support of G, then $h_{1} < {h}^{{\ast}}(x) < h_{2}$. A particular case of interest in practice is when the population consists of observations belonging to two different categories with proportions α and 1 − α having distribution functions F ₁ and F ₂. Then, the mixture distribution in (4.4) becomes

$$\displaystyle{H_{2}(x) =\alpha F_{1}(x) + (1-\alpha )F_{2}(x),}$$

which is commonly referred to as the two-component mixture model. In this case,

$$\displaystyle{{h}^{{\ast}}(x) =\alpha (x)h_{ 1}(x) + (1 -\alpha (x))h_{2}(x),}$$

where h _i (x) is the hazard rate corresponding to F _i , and

$$\displaystyle{\alpha (x) = \frac{\alpha _{1}\bar{F}_{1}(x)} {\alpha _{1}\bar{F}_{1}(x) +\alpha _{2}\bar{F}_{2}(x)};}$$

furthermore, we have

$$\displaystyle{\min (h_{1}(x),h_{2}(x)) \leq {h}^{{\ast}}(x) \leq \max (h_{ 1}(x),h_{2}(x)).}$$

For an elaborate discussion on mixtures of life distributions, one may refer to Marshall and Olkin [412].

4.2.4 Shock Models

A unit may fail because of the changes within it or due to the changes in the environment. An approach to describe the state of the system over time is through a stochastic process, and the unit is deemed to have failed when the designated process surpasses a threshold level. For units that deteriorate over time, the failure occurs as a result of accumulated shocks received over time. Let (N(t), t ≥ 0) be the number of shocks received by the unit in the time interval [0, t] and $\bar{P}_{K} = P(K > k)$, the probability that a unit survives k shocks, $k = 1,2,\ldots$. Then, the survival function of the unit is given by

$$\displaystyle{ \bar{H}_{3}(x) =\sum _{ k=0}^{\infty }[P(N(x) = k)]\bar{P}_{ K},\quad x \geq 0. }$$

(4.5)

Different models for N(t), like Poisson process, birth process and so on, have been considered in the literature. In the case of a Poisson process, (4.5) simplifies to

$$\displaystyle{ \bar{H}_{3}(x) =\sum _{ k=0}^{\infty }\bar{P}_{ K}{e}^{-\lambda x}\frac{{(\lambda x)}^{k}} {k!}. }$$

(4.6)

Attention is usually given to the problem of ascertaining whether or not a concept of ageing admits a shock model interpretation; for details, see Esary et al. [189].

4.2.5 Equilibrium Distributions

A topic that is quite useful in the analysis of the ageing phenomenon is that of equilibrium distributions. Assume that we have a set of n units. We start working with a new unit at time zero, replace it upon failure by a second unit, and so on. If the failure times X _i , $i = 1,2,\ldots,n$, of the units are independent and identically distributed, then the sequence of points (S _n ), where $S_{n} = X_{1} + X_{2} +\ldots +X_{n}$ constitute a renewal process (Cox [157]). Let F(x) be the common distribution function of X _i ’s, satisfying F(0) = 0 and μ = E(X _i ) < ∞. Upon denoting the age and remaining life of the unit in use at time T by U _T and V _T , respectively, the asymptotic distribution of U _T and V _T turn out to be

$$\displaystyle{ \bar{G}(x) {=\mu }^{-1}\int _{ x}^{\infty }\bar{F}(t)dt, }$$

(4.7)

which is called the equilibrium distribution corresponding to F. We shall denote by Z the random variable with survival function in (4.7).

For a non-negative and absolutely continuous random variable X with E(X ⁿ) < ∞, Fagiouli and Pellerey [191] extended (4.7) by defining the nth order equilibrium recursively through the survival functions

$$\displaystyle{ S_{n}(x) =\mu _{ n-1}^{-1}\int _{ x}^{\infty }S_{ n-1}(t)dt,\quad n = 1,2,3,\ldots, }$$

(4.8)

where $\mu _{n} =\int _{ 0}^{\infty }S_{n}(t)dt$. Notice that $S_{1}(x) =\bar{ F}(x)$, $S_{2}(x) =\bar{ G}(x)$ and $S_{0}(x) = \frac{f(x)} {f(0)}$, which is a survival function iff f(x) is continuous and decreasing. If Z _n is the random variable corresponding to (4.8), we also have Z ₁ = X and Z ₂ = Z according to the earlier notation. The relationships between the survival functions $\bar{F}$ and S _n , hazard rates h(x) and h _n (x), and mean residual life functions m(x) and m _n (x) of X and Z _n , respectively, are as follows (Nair and Preeth [441]):

$$\displaystyle\begin{array}{rcl} S_{n}(x)& =& \frac{E[{(X - x)}^{n-1}\vert X > x]} {E({X}^{n-1})} \bar{F}(x), {}\\ h_{n}(x)& =& {[m_{n-1}(x)]}^{-1}, {}\\ m_{n-1}(x)& =& \frac{m_{n}(x)} {1 + m_{n}^{\prime}(x)}, {}\\ h_{n-1}(x)& =& h_{n}(x) -\frac{h_{n}^{\prime}(x)} {h_{n}(x)}\quad n = 2,3,\ldots, {}\\ m_{n}(x)& =& \frac{E[{(X - x)}^{n}\vert X > x]} {nE[{(X - x)}^{n-1}\vert X > x]}. {}\\ \end{array}$$

Nair and Preeth [441] also established that the generalized Pareto distribution (see Table 1.1) is characterized by any one of the relations

$$\displaystyle\begin{array}{rcl} m_{n}(x)& =& C_{n}m(x), {}\\ h_{n}(x)& =& K_{n}h(x), {}\\ E[{(X - x)}^{n}\vert X > x]& =& A_{ n}{m}^{n}(x), {}\\ \end{array}$$

where $C_{n},\ A_{n}$ and K _n are positive constants, and the two-component exponential mixture distribution

$$\displaystyle{\bar{F}(x) =\alpha {e}^{-\lambda _{1}x} + (1-\alpha ){e}^{-\lambda _{2}x},\quad x > 0;\;\alpha \geq 0,\;0 <\lambda _{ 1} <\lambda _{2},}$$

by

$$\displaystyle{m_{n}(x) =\alpha _{1} -\alpha _{2} -\alpha _{1}\alpha _{2}h_{n}(x),}$$

where

$$\displaystyle{\alpha _{i} =\lambda _{ i}^{-1},\quad i = 1,2.}$$

Various distributional properties and reliability aspects have been discussed by Gupta [233], Nanda et al. [460], Stein and Dattero [546] and Gupta [236].

Quantile-based analysis of equilibrium distributions is straightforward. Setting x = Q(u) in (4.7), we have

$$\displaystyle{ G(Q_{X}(u)) {=\mu }^{-1}\int _{ 0}^{u}(1 - p)q(p)dp, }$$

(4.9)

where the integral

$$\displaystyle{T_{X}(u) =\int _{ 0}^{u}(1 - p)q(p)dp}$$

is the well-known total time on test transform of the random variable X. The properties and reliability implications of T _X (u) will be taken up later on in Chap. 5. Thus, from (4.9), we have

$$\displaystyle{Q_{X}(u) {=\mu }^{-1}Q_{ Z}(T_{X}(u))}$$

or

$$\displaystyle{Q_{Z}(u) =\mu Q_{X}(T_{X}^{-1}(u)).}$$

Note that T _X ^− 1(u) is a distribution function on [0, 1] since T(u) is a quantile function.

Example 4.1.

Let X be distributed as generalized Pareto with quantile function

$$\displaystyle{Q(u) = \frac{b} {a}\left [{(1 - u)}^{- \frac{a} {a+1} } - 1\right ].}$$

Then,

$$\displaystyle{T(u) =\int _{ 0}^{u}(1 - p)q(p)dp = b\left [1 - {(1 - u)}^{- \frac{1} {a+1} }\right ]}$$

and

$${\displaystyle{\mu }^{-1}T(u) = 1 - {(1 - u)}^{- \frac{1} {a+1} }.}$$

Hence, the equilibrium distribution has its quantile function as

$$\displaystyle{Q_{Z}(u) = Q[{T}^{-1}(u)] = Q\left [1 - {(1 - u)}^{-(a+1)}\right ] = \frac{b} {a}\left [{(1 - u)}^{-a} - 1\right ].}$$

We note from (4.9) that

$$\displaystyle\begin{array}{rcl} \bar{G}(Q_{X}(u))& =& 1 -{\mu }^{-1}\int _{ 0}^{u}(1 - p)q(p)dp {}\\ & =& {\mu }^{-1}\left [\mu -\int _{ 0}^{u}(1 - p)q(p)dp\right ] {}\\ & =& {\mu }^{-1}\int _{ u}^{1}(1 - p)q(p)dp. {}\\ \end{array}$$

Differentiating logarithmically, we obtain

$$\displaystyle\begin{array}{rcl} \frac{g(Q_{X}(u))} {\bar{G}(Q_{X}(u))}q(u)& =& \frac{(1 - u)q(u)} {\int _{u}^{1}(1 - p)q(p)dp} {}\\ \end{array}$$

or

$$\displaystyle\begin{array}{rcl} H_{Z}(u)& =& {[M_{X}(u)]}^{-1},{}\end{array}$$

(4.10)

thus revealing that the hazard quantile function of the equilibrium random variable is simply the reciprocal of the mean residual quantile function of the baseline distribution. Since Z _n is in the equilibrium version of Z _n − 1, we have

$$\displaystyle\begin{array}{rcl} H_{n}(u)& =& {[M_{n-1}(u)]}^{-1},\quad n = 2,3,\ldots,{}\end{array}$$

(4.11)

and so

$$\displaystyle\begin{array}{rcl} M_{n-1}(u)& =& M_{n}(u) - (1 - u)M_{n}^{\prime}(u).{}\end{array}$$

(4.12)

Equation (4.10) can also be used to derive the quantile function of Z.

Example 4.2.

The generalized lambda distribution with

$$\displaystyle{Q_{X}(u) =\lambda _{1} +\lambda _{ 2}^{-1}(u - {(1 - u)}^{\lambda _{4} }),\quad \lambda _{1} -\frac{1} {\lambda _{2}} \geq 0,}$$

has the mean residual quantile function as

$$\displaystyle{M_{X}(u) =\lambda _{ 2}^{-1}\left [\lambda _{ 4}{(1 +\lambda _{4})}^{-1}{(1 - u)}^{\lambda _{4} } + \frac{1 - u} {2} \right ].}$$

Hence,

$$\displaystyle{H_{Z}(u) = \frac{\lambda _{2}} {\lambda _{2}{(1 +\lambda _{4})}^{-1}{(1 - u)}^{\lambda _{4}} + \frac{1-u} {2} }.}$$

The corresponding quantile function can then be obtained as

$$\displaystyle{Q_{Z}(u) = \frac{1} {\lambda _{2}} \left [\frac{u} {2} + \frac{1} {1 +\lambda _{4}}(1 - {(1 - u)}^{\lambda _{4}})\right ].}$$

4.3 Classes Based on Hazard Quantile Function

4.3.1 Monotone Hazard Rate Classes

These classes of life distributions are defined by the nature of the monotonicity of the hazard function, h(x). In the sequel, we use the term increasing (decreasing) in the sense of non-decreasing (non-increasing). We say that the random variable X or its distribution has increasing hazard rate, or X is IHR in short, if and only if for all t such that $\bar{F}(t) > 0$,

$$\displaystyle{\bar{F}_{t}(x) = \frac{\bar{F}(t + x)} {\bar{F}(t)} }$$

is decreasing (increasing) in t for all x ≥ 0. This means that the residual life distribution is stochastically decreasing (increasing) in t. It is immediate that this definition also implies that X is IHR if and only if the hazard rate $h(x) = \frac{f(x)} {\bar{F}(x)}$ is increasing. Similarly, X has decreasing hazard rate iff h(x) is decreasing, and we say in this case that X is DHR. If h(x) is differentiable, it follows that X is IHR (DHR) according as h′(x) ≥ ( ≤ )0. Since

$$\displaystyle{h^{\prime}(x) = \frac{dh(x)} {dx} = \frac{dh(Q(u))} {du} \frac{du} {dQ(u)} = H^{\prime}(u) \frac{1} {q(u)}}$$

and q(u) > 0, we can present the following definition in terms of hazard quantile functions.

Definition 4.1.

A lifetime random variable is IHR (DHR) if and only if its hazard quantile function satisfies

$$\displaystyle{H^{\prime}(u) \geq (\leq )\ 0\quad \mbox{ for $0 < u < 1$.}}$$

Thus, all distributions specified in terms of F(x) that are IHR (DHR) preserve the same property when specified by Q(u) as well. We will retain the conventional nomenclature IHR (DHR) in the case of the quantile approach as well. Sometimes, it is easier to establish the IHR property by using any of the following equivalent conditions:

• (i) $H(u_{2}) \geq H(u_{1})$ for all $0 < u_{1} < u_{2} < 1,$

• (ii) The quantile function of the residual life

  $$\displaystyle{Q(u_{0} + (1 - u_{0})u) - Q(u_{0})\mbox{ is a decreasing function of $u_{0}$}.}$$

Property (i) is obvious. To prove Property (ii), we note that

$$\displaystyle\begin{array}{rcl} & & Q(u_{0} + (1 - u)u_{0}) - Q(u_{0})\text{ is decreasing} {}\\ & & \qquad \Rightarrow q(u_{0} + (1 - u_{0})u)(2 - u) - q(u_{0}) \leq 0 {}\\ & & \qquad \Rightarrow \frac{1} {(1 - u_{0})q(u_{0})} \leq \frac{1} {(1 - (u_{0} + (1 - u_{0})u))(q(u_{0} + (1 - u_{0})u))} {}\\ & & \qquad \Rightarrow H(u_{0}) \leq H(u_{0} + (1 - u_{0})u) {}\\ & & \qquad \Rightarrow \text{ IHR by (i).} {}\\ \end{array}$$

The inverse implication also holds by taking $u_{2} = u_{1} + (1 - u_{1})u$ so that $u_{2} \geq u_{1}$ for every 0 < u < 1. Taking $u_{1} = u_{0}$ and retracing the steps in the above proof, we get the required result.

Example 4.3.

The generalized exponential law with quantile function (see Table 1.1)

$$\displaystyle{Q(u) =\sigma [-\log (1 - {u}^{1/\theta })]}$$

has its hazard quantile function as

$$\displaystyle{H(u) = \frac{\theta } {\sigma (1 - u)}[{u}^{1-\frac{1} {\theta } } - u],}$$

and

$$\displaystyle{H^{\prime}(u) = \frac{\theta } {\sigma {(1 - u)}^{2}}\left [\frac{\theta -1 + u} {\theta } {u}^{-\frac{1} {\theta } } - 1\right ].}$$

Hence, H′(u) > 0 for θ > 1, H′(u) < 0 for θ < 1, and H(u) is constant for θ = 1. Thus, X is IHR for θ ≥ 1 and DHR for θ ≤ 1.

Not all hazard quantile functions are monotone in nature. It may belong to some other categories like bathtub or upside bathtub-shape, periodic, roller-coaster shape and polynomial type. These alternative forms will be discussed now briefly.

Definition 4.2.

The random variable X is said to have a bathtub-shaped hazard quantile function, or X is BT, if

$$\displaystyle{H(u) = \left \{\begin{array}{@{}l@{\quad }l@{}} H_{1}(u)\quad &u \leq u_{1} \\ c, \quad &u_{1} \leq u \leq u_{2} \\ H_{2}(u)\quad &u \geq u_{2} \end{array} \right.,}$$

where c is a constant, H ₁(u) is strictly decreasing and H ₂(u) is strictly increasing. When $u_{1},u_{2} \rightarrow 0$, X is IHR and when $u_{1},u_{2} \rightarrow 1$, X is DHR. We also say that u ₁ and u ₂ are change points of H(u). On the other hand, if H ₁(u) is strictly increasing and H ₂(u) is strictly decreasing, an upside-down bathtub-shaped (UBT) hazard quantile function results.

Often, many life distributions have only one change point in which case the following definition is more convenient.

Definition 4.3.

Assuming differentiability of H(u), X is BT (UBT) if and only if H′(u) < ( > ) 0 for u in (0, u ₀), H′(u ₀) = 0 and H′(u) > ( < ) 0 in (u ₀, 1).

Bathtub-shaped curves arise in different scenarios. In some cases, the life of a unit is affected by a mixture of defects with varying intensities. Those with serious defects at the initial stages have a high rate of failure, but as the unit functions without failure, such defects no longer persist so that the hazard function decreases and later becomes steady with almost a constant rate. Finally, when the adverse effect of age surfaces, the hazard quantile exhibits increasing tendency until the unit fails to function. Other factors such as changes in the hazard conditions due to the unit or environment, wear out of items with flaws, introduction of tests, inspection or maintenance that limits the occurrence of failures can also give rise to BT or UBT distributions. There are cases in which a proportion of items come from an IHR distribution and the remaining come from a DHR distribution, producing BT-shaped hazard for the combined set of observations, as Kao [310] has demonstrated. Table 4.1 presents the behaviour of the hazard quantile functions of standard life distributions presented earlier in Table 2.1. When failures are caused by fatigue or corrosion, X follows unimodal or UBT distributions such as lognormal, inverse Gaussian and inverted gamma, which do not have tractable quantile functions. Further details can be seen in Jiang et al. [292].

Table 4.1 Nature of hazard quantile functions of distributions in Table 2.1

Glaser [220] established a general theorem that facilitates the determination of whether X is IHR, DHR, BT or UBT. He made use of the function $\eta (x) = -\frac{f^{\prime}(x)} {f(x)}$ which is identified with

$$\displaystyle{ J(u) = \frac{q^{\prime}(u)} {{q}^{2}(u)}. }$$

(4.13)

Parzen [484] refers to J(u) as the score function, and it proves to be quite useful in classifying probability distributions according to tail length. In terms of J(u), we have the following adaptation of Glaser’s result.

Theorem 4.1.

• (a) X is IHR (DHR) according to J(u) being increasing (decreasing) for all u;

• (b) Let J(u) be BT (UBT) in the sense of Definition  4.3 . Then, if there exist a u ₀ for which J(u ₀ ) = H(u ₀), X is BT (UBT). If there is no such u ₀, then X is IHR (DHR).

Theorem 4.2.

Let

$$\displaystyle{\lim _{u\rightarrow 0} \frac{1} {q(u)} =\alpha \quad \mbox{ and}\quad \lim _{u\rightarrow 0} \frac{J(u)} {H(u)} =\beta.}$$

Then,

• (a) if J(u) is BT, X is IHR if either α = 0 or β < 1, and X is BT if either α = ∞ or β > 1;

• (b) if J(u) is UBT, X is DHR if α = ∞ or β > 1, and UBT if α = 0 or β < 1.

Remark 4.1.

Notice that, in the above two theorems, J(u) is increasing or decreasing according to

$$\displaystyle{J^{\prime}(u) = q(u)q^{\prime}(u) - {[q^{\prime}(u)]}^{2}}$$

being ≥ or ≤ 0, and

$$\displaystyle{ \frac{J(u)} {H(u)} = \frac{(1 - u)q^{\prime}(u)} {q(u)},}$$

so that the relevant quantities can be directly obtained from the quantile density function q(u).

Example 4.4.

Consider the inverse Gaussian law with probability density function

$$\displaystyle{f(x) ={ \left ( \frac{\lambda } {2\pi {x}^{3}}\right )}^{\frac{1} {2} }\exp \left [- \frac{\lambda } {{2\mu }^{2}x}{(x-\mu )}^{2}\right ],\quad x > 0;\ \lambda,\mu > 0.}$$

The distribution has no tractable quantile function. Yet, we can write

$$\displaystyle{ {[q(u)]}^{-1} ={ \left ( \frac{\lambda } {2\pi }\right )}^{\frac{1} {2} }\left \{{Q}^{-\frac{3} {2} }(u)\exp \left [- \frac{\lambda } {{2\mu }^{2}}\left (Q(u) - 2\mu + \frac{{\mu }^{2}} {Q(u)}\right )\right ]\right \},\quad 0 \leq u \leq 1, }$$

(4.14)

and

$$\displaystyle\begin{array}{rcl} J(u)& =& \frac{d} {du}{[q(u)]}^{-1} {}\\ & =&{ \left ( \frac{\lambda } {2\pi }\right )}^{\frac{1} {2} }\exp \left [-\frac{{(Q(u)-\mu )}^{2}} {{2\mu }^{2}Q(u)} \right ]\frac{q(u){Q}^{-\frac{5} {2} }(u)} {{2\mu }^{2}} \left (\frac{{3\mu }^{2}Q(u) +\lambda ({Q}^{2}(u) {-\mu }^{2})} {Q(u)} \right ) {}\\ & =& \frac{{3\mu }^{2}Q(u) +\lambda ({Q}^{2}(u) {-\mu }^{2})} {{2\mu }^{2}Q(u)}, {}\\ \end{array}$$

on using (4.14). Hence,

$$\displaystyle{J^{\prime}(u) = \frac{q(u)} {2{Q}^{3}(\mu )}(2\lambda - 3Q(u))}$$

which is increasing for $Q(u) < \frac{2\lambda } {3}$ and decreasing for $Q(u) > \frac{2\lambda } {3}$. Hence, J(u) is UBT with change point satisfying $Q(u) = \frac{2\lambda } {3}$. By Theorem 4.1 or Part (b) of Theorem 4.2, X is UBT. The same method works well for other distributions with Q(u) being not of closed form.

Gupta and Warren [249] have extended Glaser’s results to cover cases with more than one change point. Their result can be translated into quantile functions as below.

Theorem 4.3.

If Q(u) is strictly increasing and q(u) is twice differentiable and J′(u) = 0 has n solutions $0 < u_{1} < u_{2}\ldots < u_{n} < 1$, there exists at most one solution for H′(u) = 0 in [u _k−1,u _k], $k = 1,2,\ldots,n$ .

There is a vast literature on BT models including various formulations, methods of construction and applications. We will return to these issues in Chap. 8. In spite of the popularity of BT- and UBT-shaped hazard functions, there have been criticism against its indiscriminate usage and caution to the extent that they are more of a myth than reality; see, for example, Tabot [558] and Wong [586]. Klutke et al. [343] pointed out that a bimodal density function as a mixture of subpopulations does not yield a decreasing hazard function early in life. This means that a bathtub hazard rate cannot accommodate this feature of early failures. Further, if the hazard rate is decreasing in an interval, the density must also decrease in that interval. In a series of papers, Wong [587–589] advocated the concept of roller-coaster-shaped hazard functions citing the following physical characteristics for the hazard function to take such a shape. The shape is initially generated by basic failure mechanisms which lead to decreasing hazard rate and then humps are created by changing hazard conditions, wear out, distribution of flawed items, etc. Roller-coaster curves thus exhibit alternate monotonicities repeatedly.

Definition 4.4.

If there exist points $0 < u_{1} < u_{2}\ldots < u_{k} < 1$ such that in the interval $[u_{k-1},u_{k}]$, 1 ≤ k ≤ n + 1, u ₀ = 0, $u_{n+1} = 1$, H(u) is strictly monotone and it has opposite monotonicity in any two adjacent such intervals, we say that X has roller-coaster-shaped hazard quantile function with change points $u_{1},u_{2},\ldots,u_{k}$.

Another typical criterion of interest in this context is that of periodic hazard rate. A hazard function h(x) is periodic if

$$\displaystyle{B(x,t) = B(x + nc,t)}$$

n ≥ 0 is an integer and C ≥ 0, where

$$\displaystyle{B(x,t) = \frac{F(x + t) - F(x)} {1 - F(x)}.}$$

Prakasa Rao [496] has shown that distributions with periodic hazard rates will be of the form

$$\displaystyle{\bar{F}(x) = p(x){e}^{-\alpha x},\quad x \geq 0;\,\alpha > 0,}$$

or exponential, where p( ⋅) is a periodic function with period c and support contained in the set (nc, n ≥ 0). This follows from the fact that the hazard function is either a constant (in which case the definition is trivially satisfied since X is exponential) or has the form

$$\displaystyle{h(x) = \frac{\alpha p(x) - p^{\prime}(x)} {p(x)} }$$

along with the fact that p(x) being periodic, p′(x) is also periodic. It can also be seen that (Chukova and Dimitrov [148] and Chukova et al. [149]) a non-negative random variable with continuous density function has periodic hazard rate if and only if it has almost lack of memory property. The almost lack of memory property means that there exists a sequence of distinct constants (a _n ) such that

$$\displaystyle{P(X \geq b + x\vert X \geq b) = P(X \geq x)}$$

holds for any b = a _n , $n = 1,2,3,\ldots$ and all x ≥ 0. Castillo and Sieworek [130] have considered the reliability of computing systems and showed that hard disk failures seem to follow the work load. The influence of this work load can be accounted for by addition of a periodic hazard rate. More details are present in Chukova and Dimitrov [148], Boyan [119], Dimitrov et al. [179] and Tanguy [561].

Apart from providing monotone hazard rates, the IHR and DHR classes of distributions possess some other important properties, which are as follows:

• 1. If X ₁ and X ₂ are IHR, their convolution is IHR, while DHR does not preserve convolution;

• 2. It is not true that a coherent system of independent IHR components is necessarily IHR. The DHR class is also not closed under the formation of coherent systems. A parallel system of independent and identical IHR units is IHR while a series system of IHR units, not necessarily identical, is also IHR;

• 3. The operation of formation of mixtures is preserved under DHR class and is not preserved for the IHR class;

• 4. The harmonic mean of two IHR survival probabilities is an IHR survival probability;

• 5. If X is IHR (DHR), then $\log \bar{F}(x)$ is concave (convex);

• 6. If X is IHR (DHR) and ξ _p is the pth percentile, then

  $$\displaystyle\begin{array}{rcl} \begin{array}{lll} \bar{F}(x)& \geq (\leq )\ {e}^{-\alpha x},&x \leq \xi _{p} \\ & \leq (\geq )\ {e}^{-\alpha x},&x \geq \xi _{p} \end{array},& & {}\\ \end{array}$$

  where $\alpha = -\frac{\log (1-p)} {\xi _{p}}$;

• 7. If X is IHR, then

  $$\displaystyle{\bar{F}(x) \geq \left \{\begin{array}{@{}l@{\quad }l@{}} {e}^{-\frac{x} {\mu } },\quad &t <\mu \\ 0, \quad &t \geq \mu \end{array} \right.,}$$

  or equivalently

  $$\displaystyle{Q(u) \geq -\mu \log (1 - u)\text{ for }t <\mu;}$$

• 8. If X is IHR, then

  $$\displaystyle{-\mu \log (1 - u) \leq \xi _{u} \leq -\mu \frac{\log (1 - u)} {u},\quad u \leq 1 - {e}^{-1},}$$

  and

  $$\displaystyle{\mu \leq \xi _{u} \leq -\mu \frac{\log (1 - u)} {u},\quad \mu \geq 1 - {e}^{-1};}$$

• 9. If X is DHR, then

  $$\displaystyle\begin{array}{rcl} Q(u)& \leq &-\mu \log (1 - u),\quad Q(u) \leq \mu, {}\\ & \leq & \frac{\mu {e}^{-1}} {(1 - u)},\quad Q(u) \geq \mu; {}\\ \end{array}$$

• 10. If X ₁ and X ₂ are IHR with hazard rates h ₁(x) and h ₂(x), then the hazard rate of the convolution h _c (x) satisfies $h_{c}(x) \leq \min (h_{1}(x),h_{2}(x))$;

• 11. The quantile density function of an IHR random variable is increasing;

• 12. We define the kth spacing between order statistics $X_{1:n},\ldots,X_{n:n}$ by $X_{k:n} - X_{k-1:n}$. The order statistics of IHR distributions are IHR, but the spacings of IHR distributions are not IHR. In the case of DHR class, the spacings are DHR, while order statistics are not;

• 13. In life testing experiments, some units may not fail at all during the course of the test. Hence, one cannot calculate the sample mean life for estimating μ. If M is the median, using Property 9 above, the mean can then be estimated as

  $$\displaystyle{\frac{M} {2\log 2} \leq \mu \leq \frac{M} {\log 2};}$$

• 14. If X ₁ and X ₂ are identically distributed and IHR, then (Ahmad [21])

  $$\displaystyle{{2}^{\frac{(r+2)(r-1)} {2} }E{(\min (X_{1},X_{2}))}^{r} \geq r{!\mu }^{r},\quad r \geq 2,}$$

  and

  $$\displaystyle{E{(\min (X_{1},X_{2}))}^{2r+2} \geq \binom{2r + 2}{2r + 1}{\left (\frac{1} {2}\right )}^{2r+2}{(\mu _{ r+1}^{\prime})}^{2};}$$

• 15. If X is IHR (DHR), its residual life X _t is also IHR (DHR);

• 16. If X is IHR (DHR), Z is also IHR (DHR). The converse is true if and only if the ratio of their densities $\frac{f(x)} {g(x)}$ is increasing (decreasing).

Results 1–12 are discussed in Barlow and Proschan [70] while 15 and 16 are presented in Gupta and Kirmani [242].

The equilibrium distribution discussed in Sect. 4.2 is a particular case of a more general class of distributions called weighted distributions. The random variable Y with density function

$$\displaystyle{g(x) = \frac{w(x)f(x)} {Ew(X)},\quad w(x) > 0,\;Ew(X) < \infty,}$$

specifies the weighted distribution corresponding to the random variable X. Then,

$$\displaystyle{h_{Y }(x) = \frac{w(x)} {E[w(X)\vert X > x]}h_{X}(x).}$$

The equilibrium distribution results as a special case when $w(x) = \frac{1} {h(x)}$. For various identities connecting reliability functions of X and Y and conditions on the distribution of X for preserving DHR/IHR property of X, one may refer to Jain et al. [290], Gupta and Kirmani [242], Oluyede [473], Bartoszewicz and Skolimowska [77] and Misra et al. [417]. Returning to the quantile approach,

$$\displaystyle{G(x) = \frac{\int _{0}^{x}w(x)f(x)dx} {\mu _{w}},\quad \mbox{ with }\mu _{w} = E[w(X)],}$$

becomes

$$\displaystyle{{F}^{{\ast}}(u) = G(Q(u)) = \frac{\int _{0}^{u}W(p)dp} {\mu _{W}},\quad \mbox{ with }W(p) = w(Q(p)),}$$

which is a distribution function on (0, 1).

Blazej [107] has shown that if X is IHR and C(u) is an increasing positive function on (0, 1) such that

$$\displaystyle{\int _{0}^{1}\frac{C(u)du} {1 - u} = +\infty,}$$

then Y is IHR. For a choice of C(u), one can choose it as a constant, a positive increasing function on (0, 1), or a positive decreasing function ϕ such that ϕ(x) > a > 0.

Lariviere and Porteus [375] refer to the ratio $t(x) = \frac{xf(x)} {\bar{F}(x)}$ as a generalized hazard rate and if t(x) is non-decreasing, X is said to be increasing generalized hazard rate (IGHR). Lariviere [374] has shown that X is IGHR is equivalent to logX being IHR. Further, if logX is IHR (IGHR), then α + βlogX (α X ^β) is IHR (IGHR). The concept of generalized hazard rate is primarily intended for use in operations management. Occasionally, the above results on IGHR becomes handy in verifying whether X is IHR.

4.3.2 Increasing Hazard Rate(2)

The notion of stochastic dominance plays a role in defining certain ageing classes. If X ₁ and X ₂ are two lifetime random variables with distribution functions F ₁ and F ₂, respectively, then X ₁ has stochastic dominance over X ₂ of the first order, D ₁, if $F_{1}(x) \leq F_{2}(x)$. The dominance of order 2, D ₂, is defined as

$$\displaystyle{\int _{0}^{x}F_{ 1}(t)dt \leq \int _{0}^{x}F_{ 2}(t)dt}$$

and $\bar{D}_{2}$ as

$$\displaystyle{\int _{x}^{\infty }\bar{F}_{ 1}(t)dt \geq \int _{x}^{\infty }\bar{F}_{ 2}(t)dt,}$$

while the third order stochastic dominance D ₃ and $\bar{D}_{3}$ are

$$\displaystyle{\int _{0}^{y}\int _{ 0}^{x}F_{ 1}(t)dtdx \leq \int _{0}^{y}\int _{ 0}^{x}F_{ 2}(t)dtdx}$$

and

$$\displaystyle{\int _{y}^{\infty }\int _{ x}^{\infty }\bar{F}_{ 1}(t)dtdx \geq \int _{y}^{\infty }\int _{ x}^{\infty }\bar{F}_{ 2}(t)dtdx,}$$

respectively. Implications among these orders are as follows:

$$\displaystyle{D_{1} \Rightarrow D_{2}(\bar{D}_{2}) \Rightarrow D_{3}(\bar{D}_{3}).}$$

Deshpande et al. [172] defined increasing hazard rate of order 2 (IHR(2)) by requiring the residual life $X_{t_{1}}$ to have stochastic dominance D ₂ over $X_{t_{2}}$. In other words, X is IHR(2) if and only if for every fixed x ≥ 0, one of the following conditions are satisfied:

• (a) $\int _{0}^{x}\frac{\bar{F}(t+s)} {\bar{F}(t)} ds\mbox{ is non-decreasing in $s$};$

• (b) $\int _{0}^{x}\frac{\bar{F}(t+s)} {\bar{F}(t)} dt \leq \int _{0}^{x}\frac{\bar{F}(t+v)} {\bar{F}(t)} dt\quad \forall x \geq 0,\;s \geq v.$

Definition 4.5.

A lifetime random variable X is IHR(2) if and only if

$$\displaystyle{ \int _{0}^{u}[Q(t + (1 - t)v) - Q(t)]dt \geq \int _{ 0}^{u}Q[(t + (1 - t)s) - Q(t)]dt }$$

(4.15)

for all u ≥ 0 and t ≤ s.

When X is IFR, we have

$$\displaystyle{Q(t + (1 - t)v) - Q(t) \geq Q(t + (1 - t)s) - Q(t)}$$

which implies (4.15) on integrating the last inequality over (0, u). Hence, IHR ⇒ IHR(2). Since $g_{1}(x) \geq g_{2}(x)$ does not always imply $\frac{dg_{1}} {dx} \geq \frac{dg_{2}} {dx}$, IHR(2) need not imply IHR, thus proving IHR(2) contains IHR. However, this class does not seem to have received much attention in practice.

4.3.3 New Better Than Used in Hazard Rate

This concept uses the idea that at the initial age the hazard rate will be less than that of a used one, indicating positive ageing. We say that X is new better than used in hazard rate (NBUHR) if h(0) ≤ h(x) for x ≥ 0. The dual class is new worse than used in hazard rate (NWUHR) if h(0) ≥ h(x) for x ≥ 0. When the term failure rate is used instead of hazard rate, the abbreviation becomes NBUFR, which is often used in the literature (Loh [404]).

Definition 4.6.

A lifetime X is new better (worse) than used in hazard rate if and only if H(0) ≤ ( ≥ ) H(u) for u ≥ 0.

An associated concept is that of new better than used in hazard rate average (NBUHRA). Let F(x) be such that F(0) = 0 and

$$\displaystyle{\frac{\log \bar{F}(x)} {x} \leq \lim _{x\rightarrow 0}\frac{\log \bar{F}(x)} {x}.}$$

Then, X is NBUHRA. If $\bar{F}(x)$ is continuously differentiable over (0, ε) for some ε > 0, then the above condition is also equivalent to

$$\displaystyle{ h(0) \leq \frac{1} {x}\int _{0}^{x}h(t)dt,\quad x \geq 0. }$$

(4.16)

In this case, the hazard rate of a new component is less than its average hazard rate in (0, x) for all x ≥ 0 (Loh [404]). Further, X is new worse than used in hazard rate average (NWUHRA) if the inequality in (4.16) is reversed. The NBUHRA and NWUHRA classes are also denoted by NBUFRA and NWUFRA in the literature. We now rewrite the definitions of the classes by making use of (4.16) in terms of hazard quantile function.

Definition 4.7.

We say that X is NBUHRA (NWUHRA) if and only if the hazard quantile function H(u) satisfies

$$\displaystyle{ H(0) \leq (\geq )\ \frac{-\log (1 - u)} {Q(u)} \quad \text{for all }u. }$$

(4.17)

Definition 4.7 can also be seen as

$$\displaystyle{Q(u) \leq - \frac{1} {H(0)}\log (1 - u)}$$

meaning that the quantile function of X is less than that of the exponential distribution with the same hazard rate as X. In the first order stochastic dominance mentioned above, the lifetime X is worse than the exponential and hence X ages positively if it is NBUHRA. The process of averaging visible in (4.16) is also implicit in (4.17) if one rewrites it as

$$\displaystyle{H(0) \leq (\geq )\ \frac{\int _{0}^{u}H(p)q(p)dp} {\int _{0}^{u}q(p)dp}.}$$

The right side acts as a weighted average with weight $\frac{q(u)} {Q(u)}$. Evidently,

$$\displaystyle{\text{IHR } \Rightarrow \text{NBUHR}}$$

and furthermore

$$\displaystyle\begin{array}{rcl} \text{NBUHR}& \Rightarrow & H(0) \leq H(u) \Rightarrow \int _{0}^{u}H(0)q(p)dp \leq \int _{ 0}^{u}H(p)q(p)dp {}\\ & \Rightarrow & \text{NBUHRA}, {}\\ \end{array}$$

but not conversely. A closely related measure of positive ageing, given in Bryson and Siddiqui [122], based on the interval-average hazard rate defined by

$$\displaystyle{A(t,s) = \frac{1} {t}\int _{s}^{s+t}h(x)dx,}$$

is increasing interval average hazard rate when

$$\displaystyle{A(t_{2},s) \geq A(t_{1},s),\quad t_{2} \geq t_{1} \geq 0,\;s \geq 0.}$$

They have shown that this criterion is equivalent to IHR. The properties of the above two classes have been discussed in Abouammoh and Ahmed [5], Gohout and Kunhert [222] and El-Bassiouny et al. [185]. It has been shown that

• 1. NBUHR is not preserved under convolution,

• 2. NBUHR class is closed under the formation of mixtures,

• 3. NBUHR class is closed under formation of coherent systems with independent components,

• 4. $\mu _{r}^{\prime} \geq (\leq )\frac{f(0)\mu _{r+1}^{\prime}} {(r+1)}$ according to X being NBUHR (NWUHR)

4.3.4 Stochastically Increasing Hazard Rates

The monotonicity of hazard rates can also be evaluated in random intervals of time giving a further extension of IHR and DHR concepts.

Definition 4.8.

Let $X_{0} = 0,X_{1},\ldots,X_{k},\ldots$ be a sequence of independent and identically distributed exponential random variables each with mean μ, and Y be independent of X _k . Then, Y is said to have stochastically increasing hazard rate, or Y is SIHR, if and only if (Singh and Deshpande [542])

$$\displaystyle{P\left [\sum _{0}^{k}X_{ i} \leq Y <\sum _{ 0}^{k+1}X_{ i}\vert Y \geq \sum _{0}^{k}X_{ i}\right ] \geq P\left [\sum _{0}^{k-1}X_{ i} \leq Y <\sum _{ 0}^{k}X_{ i}\vert Y \leq \sum _{0}^{k-1}X_{ i}\right ].}$$

This means that the conditional probability of a unit with lifetime Y will not fail before the random time $\sum _{1}^{k}X_{i}$ given that it has not failed in $\sum _{0}^{k-1}X_{i}$, is decreasing in $k = 1,2,3,\ldots$. The dual class SDHR is defined similarly. It may be noted that

$$\displaystyle{\text{IHR } \Rightarrow \text{SIHR}}$$

and that both SIHR and SDHR hold if and only if Y is exponential. This concept has not been used much in reliability analysis.

4.3.5 Increasing Hazard Rate Average

Introduced by Birnbaum et al. [102], the increasing hazard rate average (IHRA) class and its dual decreasing hazard rate average (DHRA) class are among the basic classes of life distributions. A lifetime X is IHRA (DHRA) if and only if

$$\displaystyle{-\frac{\log \bar{F}(x)} {x} \text{ is increasing (decreasing) in }x > 0.}$$

Since $-\log \bar{F}(x) =\int _{ 0}^{x}h(t)dt$, X is IHRA means that the average hazard rate in (0, x) defined by

$$\displaystyle{\frac{1} {x}\int _{0}^{x}h(t)dt\quad \text{ is increasing.}}$$

A real valued function ϕ(x) on [0, ∞) is star-shaped if ϕ(0) = 0 and $\frac{\phi (x)} {x}$ is increasing in x > 0. Hence, X is IHRA if and only if $\log \bar{F}(x)$ is star-shaped.

Definition 4.9.

We say that X is IHRA (DHRA) if and only if $\frac{Q(u)} {-\log (1-u)}$ is decreasing (increasing) in 0 < u < 1.

Theorem 4.4.

The following conditions are equivalent:

• (i) X is IHRA;

• (ii) $\frac{\int _{0}^{u}H(p)q(p)dp} {\int _{0}^{u}q(p)dp}$ is increasing;

• (iii) $H(u) \geq \frac{Z(u)} {Q(u)}$, with $Z(u) = -\log (1 - u)$ .

Proof.

• (i)  ⇔ (ii)

  $$\displaystyle\begin{array}{rcl} \mbox{ $X$ is IHRA }& \Leftrightarrow &-\frac{\log (1 - u)} {Q(u)} \text{ is increasing} {}\\ & \Leftrightarrow &\mbox{ (ii), since $ -\log (1 - u) =\int _{ 0}^{u}H(p)q(p)dp$.} {}\\ \end{array}$$

• (ii)  ⇔ (iii). From (ii), we have

  $$\displaystyle\begin{array}{rcl} \frac{\int _{0}^{u}H(p)q(p)dp} {\int _{0}^{u}q(p)dp} \text{ is increasing}& \Leftrightarrow & Q(u)H(u)q(u) - q(u)\int _{0}^{u}H(p)q(p)dp \geq 0 {}\\ & \Leftrightarrow & \frac{Q(u)} {1 - u} - q(u)Z(u) \geq 0 {}\\ & \Leftrightarrow & \frac{1} {(1 - u)q(u)} \geq \frac{Z(u)} {Q(u)} {}\\ & \Leftrightarrow & H(u) \geq \frac{Z(u)} {Q(u)}. {}\\ \end{array}$$

We now list some key properties of the IHRA and DHRA classes.

• 1. IHR ⇒ IHRA. The converse need not be true. Marshall and Olkin [412] have shown that

  $$\displaystyle{\bar{F}(x) = {e}^{-\lambda _{1}x} + {e}^{-\lambda _{2}x} - {e}^{-(\lambda _{1}+\lambda _{2})x}}$$

  is IHRA, but X is UBT.

• 2. If X is IHRA, then $\frac{\bar{F}(x)} {\bar{F}_{Z}(x)}$ is decreasing, where as before Z is the equilibrium random variable.

• 3. If X belongs to the one-parameter exponential family of distributions and E(Z) = E(X), then X has exponential distribution.

• 4. $h_{Z}(x) = c\,h_{X}(x)$, where c > 0 is a constant, if and only if X follows generalized Pareto distribution.

• 5. When X is IHRA, its survival function $\bar{F}(x)$ can cross the survival function of any exponential at most once and only from above.

• 6. An IHRA distribution has finite moments of all orders.

• 7. The class of IHRA distributions is closed under the formation of coherent systems. It is the smallest class containing the exponential law, while DHRA class is not closed. Both IHRA and DHRA preserve formation of series systems.

• 8. The IHRA class is closed under convolution and its dual DHRA class does not possess such a property.

• 9. Mixtures of IHRA distributions are not necessarily IHRA, while if each F(x; θ) is DHRA, then the mixture is also DHRA.

• 10. Every IHRA distribution can be obtained as a limit in distribution of a sequence of coherent systems of components having exponential or degenerate distributions.

• 11. IHRA distributions arise when shocks occur according to a Poisson process in time, each independently causing random damage to the unit. The unit fails when accumulated damages exceed a threshold level.

• 12. We have

  $$\displaystyle\begin{array}{rcl} \mu _{r}^{\prime}& \leq & (\geq )\ \varGamma {(r + 1)\mu }^{r},\quad 0 < r < 1 {}\\ \mu _{r}^{\prime}& \geq & (\leq )\ \varGamma {(r + 1)\mu }^{r},\quad 1 < r < \infty. {}\\ \end{array}$$

• 13. We have

  $$\displaystyle{\bar{F}(x)\left \{\begin{array}{@{}l@{\quad }l@{}} \geq (\leq )\ {e}^{-\alpha x},\quad &0 < x <\xi p \\ \leq (\geq )\ {e}^{-\alpha x},\quad &x >\xi p,\;\alpha = -\frac{1} {\xi p}\log (1 - p). \end{array} \right.}$$

• 14. The coefficient of variation is ≤ ( ≥ )1 according to X being IHRA (DHRA).

• 15. $E(X_{1}^{r+1}) \geq E{[\min (\frac{X_{1}} {\alpha }, \frac{X_{2}} {1-\alpha })]}^{r+1}$, where X ₁ and X ₂ are independent and identically distributed IHRA variables (with r being an integer); see Ahmad and Mugadi [26].

• 16. For fixed x > 0, when X is IHRA,

  $$\displaystyle{\bar{F}(x) \leq \left \{\begin{array}{@{}l@{\quad }l@{}} 1 \quad &,\ x \leq \mu \\ {e}^{-\omega x } \quad &,\ x >\mu \end{array} \right.,}$$

  where ω is a function of x satisfying $1-\omega \mu = {e}^{-\omega x}$.

• 17. When $X_{1},X_{2},\ldots,X_{n}$ are iid IHRA, for all integers r ≥ 0, k ≥ 2, we have (Ahmad and Mugadi [26])

  $$\displaystyle{E{(\min (X_{1},X_{2},\ldots,X_{n}))}^{r} \geq \frac{\mu _{r+1}^{\prime}} {r + 1}.}$$

• 18. Let X be IHRA (DHRA). Then, ${[ \frac{\mu _{r}^{\prime}} {\varGamma (r+1)}]}^{\frac{1} {r} }$ is decreasing (increasing) in r ≥ 0.

• 19. If X is IHRA (DHRA) and C(u) is an increasing (decreasing) function on (0, 1), then $G(x) = \frac{\int _{0}^{x}w(x)f(x)} {\mu _{x}}$ is IHRA (DHRA) (Blazej [107]).

Proofs and further details of many of the above results can be found in Barlow and Proschan [69]. For results concerning weighted distributions, see Misra et al. [417], Bartoszewicz and Skolimowska [77] and Oluyede [473].

4.3.6 Decreasing Mean Time to Failure

The IHR and IHRA arise in evolving maintenance strategies in reliability engineering. In order that a unit functions satisfactorily without failures or disruption, reliability engineers adopt several types of maintenance strategies. These strategies spell out schemes of replacement before failure. One such method is to resort to an age-replacement policy in which a unit is replaced either when it fails or at an age T whichever is earlier. The number of failures N(x) in (0, x) with no planned replacements is a renewal process and N _A (x, T) and the number of in-service failures under the age-replacement policy is also a renewal process. If $\bar{F}(x)$ is the survival function of X _i , the length of time between the (i − 1)th and ith failures in the process N(x), the distribution of X _i, A , the length of time between the (i − 1)th and ith failures in N _A (x, T) for fixed T > 0, is specified by

$$\displaystyle{ S_{T}(x) = {[\bar{F}(T)]}^{n}\bar{F}(x - nT),\quad nT \leq x < (n + 1)T,\;n = 0,1,2,\ldots }$$

(4.18)

A yardstick for the effectiveness of the strategy is to study the properties of the mean time to failure (MTTF) derived as the mean of (4.18), viz.,

$$\displaystyle{ M(T) = \frac{1} {F(T)}\int _{0}^{T}\bar{F}(t)dt. }$$

(4.19)

Equation (4.19) makes it clear that the behaviour of M(T) is associated with ageing properties of F. Thus, the class of distributions for which M(T) is decreasing (DMTTF) and increasing (IMTTF) have been studied in literature by many authors including Klefsjö [334], Knopik [344, 345] and Asha and Nair [39]. We can rewrite (4.19) as

$$\displaystyle{ \mu (u) = M(Q(u)) = {u}^{-1}\int _{ 0}^{u}(1 - p)q(p)dp }$$

(4.20)

which gives the average time to failure at the 100(1 − u)% point of the distribution.

Definition 4.10.

We say that S _T (x) is decreasing (increasing) MTTF according to μ ′(u) ≤ ( ≥ ) 0. Then, we have the identities

$$\displaystyle\begin{array}{rcl} \mu (u)& =& {u}^{-1}[\mu -(1 - u)R(u)], {}\\ u\mu ^{\prime}(u) +\mu (u)& =& {[H(u)]}^{-1}. {}\\ \end{array}$$

Thus, μ(u) is increasing or decreasing according to $\mu (u) \geq {(H(u))}^{-1}$ or $\mu (u) \leq {[H(u)]}^{-1}$, for all u in (0, 1). The MTTF is BT (UBT) when H(u)μ(u) = 1 and H′(u ₀) ≤ ( ≥ )0 at u ₀, where u ₀ is the solution of the equation H(u)μ(u) = 1. Li and Xu [393] have shown that

$$\displaystyle{\text{IHR} \Rightarrow \text{IHRA} \Rightarrow \text{DMTTF}.}$$

Asha and Nair [39] have further proved that if μ(u) is decreasing and concave, then X is IHR and hence IHRA. Knopik [344, 345] has established the following closure properties:

• 1. If the lifetimes of the components are independent with absolutely continuous distributions which are DMTTF, then any parallel system is also DMTTF, and moreover if the components are identically distributed also, then a series system of DMTTF components is DMTTF;

• 2. The DMTTF family is closed under convolution and weak convergence of distributions.

Li and Li [398] introduced the IHRA* t ₀ (DHRA* t ₀) classes which imply that the average hazard rate begins to increase (decrease) after t ₀.

Definition 4.11.

A random life X is IHRA* t ₀, for all x ≥ t ₀ > 0 and all $\frac{t_{0}} {x} \leq b < 1$, if and only if $\bar{F}(bx) \geq \bar{ {F}}^{b}(x)$.

This class of IHRA* t ₀ distributions satisfies the following properties:

• 1. IHRA ⇒ IHRA* t ₀;

• 2. Let F be a life distribution with strictly increasing hazard rate. Denote the cumulative hazard rate of F by $C_{F}(x) =\int _{ 0}^{x}h(t)dt$ and

  $$\displaystyle{{C}^{{\ast}}(x) = \left \{\begin{array}{@{}l@{\quad }l@{}} C_{F}(x) \quad &,\ 0 \leq x \leq t_{0} \\ C_{G}(x) - C_{G}(t_{0}) + C_{F}(t_{0})\quad &,\ x \geq t_{0}\end{array} \right..}$$

  If $C_{F}(t_{0}) \leq C_{G}(t_{0})$, then the life distribution determined by C ^∗ is IHRA* t ₀, but not IHRA. For example, take $\bar{F}(x) = {a}^{{x}^{1/2} }$, 0 < a < 1, $\bar{G}(x) = {e}^{-{x}^{2} }$, x ≥ 0, $t_{0}^{3/2} \geq -\log a$;

• 3. If $\bar{F}_{i}$ is the survival function of an IHRA ∗ t _i unit, $i = 1,2,\ldots,n$, of a system, then the coherent system is IHRA ∗ t ₀, where $t_{0} = \max _{1\leq i\leq n}t_{i}$.

4.4 Classes Based on Residual Quantile Function

In this section, we discuss various classes of life distributions arising from the monotonic nature of the mean, variance and percentile residual functions. As in the case of the hazard rate notions, the classes are identical irrespective of whether the definitions based on distribution functions or quantile functions are implemented.

4.4.1 Decreasing Mean Residual Life Class

For defining increasing hazard function classes, we utilized the fact that the quantile function of the residual life is decreasing in u ₀. A weaker class can be obtained if we consider instead the monotonicity of the mean of the distribution. Thus, we have the decreasing (increasing) mean residual life DMRL (IMRL) class if m(x) is a decreasing (increasing) function in x > 0.

Definition 4.12.

A random variable X is said to have decreasing (increasing) mean residual quantile function if

$$\displaystyle{M(u_{1}) \leq (\geq )M(u_{2})\quad \text{ for }0 \leq u_{2} \leq u_{1} < 1,}$$

or equivalently

$$\displaystyle{\int _{0}^{1}[Q(u + (1 - u)p) - Q(u)]dp}$$

is a decreasing (increasing) function of u.

Setting $v = u + (1 - u)p$, we see that

$$\displaystyle{\int _{0}^{1}[Q(u + (1 - u)p) - Q(u)]dp = {(1 - u)}^{-1}\int _{ u}^{1}Q(v)dv - Q(u) = M(u).}$$

If M(u) is differentiable, then X is DMRL (IMRL) according to M′(u) ≤ ( ≥ )0.

Example 4.5.

Let

$$\displaystyle{Q(u) =\sigma (1 - {(1 - u)}^{\frac{1} {\alpha } }),\quad \sigma,\alpha > 0.}$$

Then, we have

$$\displaystyle{M(u) = \frac{\sigma } {\alpha +1}{(1 - u)}^{\frac{1} {\alpha } }}$$

so that

$$\displaystyle{M^{\prime}(u) = - \frac{\sigma } {\alpha (\alpha +1)}{(1 - u)}^{\frac{1} {\alpha } -1} < 0.}$$

Hence, X is DMRL.

Abouammoh and El-Neweihi [4], Abouammoh et al. [6], Gupta and Kirmani [242], Abu-Youssef [16] and Ahmad and Mugadi [26] have all discussed various properties of the classes of distributions generated by monotonic mean residual life function. Some of these properties are as follows:

• 1. If X is IHR (DHR), then X is DMRL (IMRL). This follows from the implications

  $$\displaystyle\begin{array}{rcl} X\text{ is IHR}& \Rightarrow & Z\text{ is IHR} \Rightarrow H_{z}(u) = \frac{1} {M_{X}(u)}\text{ is increasing} {}\\ & \Rightarrow & M_{X}(u)\text{ is decreasing.} {}\\ \end{array}$$

  The converse need not be true. However, if X is DMRL and m(x) is convex, then X is IHR.

• 2. X is DMRL if and only if $h_{X}(x) \leq h_{Z}(x)$ or $H_{X}(u) \leq H_{Z}(u)$. We note from (2.37) that

  $$\displaystyle\begin{array}{rcl} H_{X}(u) \leq H_{Z}(u)& \Leftrightarrow & H_{X}(u) \leq \frac{1} {M_{X}(u)} {}\\ & \Leftrightarrow & H_{X}(u)M_{X}(u) \leq 1 {}\\ & \Leftrightarrow & 1 - H_{X}(u)(1 - u)M_{X}^{\prime}(u) \leq 1 {}\\ & \Leftrightarrow & M_{X}^{\prime}(u) < 0. {}\\ \end{array}$$

• 3. From Property 1, it follows that X is DMRL if and only if Z is IHR.

• 4. X is DMRL (IMRL) if and only if Eϕ(X − x | X > x) is decreasing (increasing) for all convex increasing functions ϕ.

• 5. The mixture of IMRL distributions is IMRL, provided the mixture has a finite mean. Generally, DMRL distributions are not closed under the formation of mixtures. However, for the mixture of non-crossing life distributions, IMRL class is closed.

• 6. Both IMRL and DMRL classes are not closed under the formation of coherent systems.

• 7. The convolution of both IMRL and DMRL distributions need not be of the same class.

• 8. We have

  $$\displaystyle{(r + 1)E[X_{1}{(\min X_{1},X_{2})}^{r}] \geq (\leq )(r + 2)\nu _{ 2}^{(r+1)},}$$

  where X ₁ and X ₂ are independent and identically distributed, $\nu _{r} = E{(\min X_{1},X_{2})}^{r}$, and also

  $$\displaystyle{\nu _{2} \geq (\leq )\ \frac{{\mu }^{2}} {2}}$$

  when F is DMRL (IMRL).

• 9. $M_{Z}(u) \leq (\geq )M_{X}(u)$ if and only if X is DMRL.

• 10. If θ(u) is increasing and X is DMRL, then X is IHR. Recall the definition of

  $$\displaystyle{\theta (u) = \frac{1} {1 - u}\int _{u}^{1}Q(p)dp}$$

  as the mean quantile function. Then, upon differentiation, we find

  $$\displaystyle{(1 - u)\theta ^{\prime}(u) -\theta (u) = -Q(u)}$$

  and

  $$\displaystyle{M(u) + Q(u) =\theta (u),}$$

  which together yields

  $$\displaystyle{ \frac{\theta ^{\prime}(u)} {M(u)} = \frac{1} {1 - u} = q(u)H(u).}$$

• 11. Let

  $$\displaystyle{A_{1}(x,y) = \frac{1} {\bar{F}(x)}\int _{x}^{y}w(t)f(t)dt,\quad x < y.}$$

  If X is DMRL (IMRL), A ₁(x, y) is increasing (decreasing) on the support of X, and log convex (log concave) for x satisfying f(x) > 0, then the weighted random variable X _W is DMRL (IMRL) (Misra et al. [417]).

• 12. If m(x) is strictly convex (concave) on [0, ∞) and decreasing (increasing) for x ≥ 0, then h(x) is strictly increasing (decreasing) on [0, ∞) (Kupka and Loo [363]).

• 13. A random variable with E(X ⁿ) < ∞, $n = 1,2,\ldots$, is generalized Pareto if and only if $M_{n}(u) = C_{n}M(u)$, where M _n (u) is as defined in (4.12).

• 14. IHRA does not imply DMRL.

When the mean residual life declines in the interval (0, t ₀) and thereafter never greater than what it was at age t ₀ (Kulasekera and Park [356]), the class of distributions is called BMRL-t ₀ (better mean residual life at age t ₀).

Bryson and Siddiqui [122] introduced net decreasing mean residual lifetime of X if and only if m(x) ≤ m(0) for all x ≥ 0. This translates into the following definition.

Definition 4.13.

We say that X has net decreasing mean residual lifetime (NDMRL) if and only if M(u) ≤ M(0) for 0 ≤ u < 1.

The NDMRL has the following implications:

$$\displaystyle\begin{array}{rcl} \text{IHR }& \Rightarrow &\text{ DMRL } \Rightarrow \text{ NDMRL}, {}\\ \text{IHR }& \Rightarrow &\text{ IHRA } \Rightarrow \text{ NDMRL.} {}\\ \end{array}$$

The dual class is defined by reversing the inequality and the implications among corresponding negative ageing concepts follow. Another criterion based on mean residual quantile function takes the harmonic averages in (0, x). A distribution is decreasing mean residual life in harmonic average (DMRLHA) if and only if (Deshpande et al. [172])

$$\displaystyle{{\left [\frac{1} {x}\int _{0}^{x} \frac{dt} {m(t)}\right ]}^{-1}\mbox{ is decreasing in $x$.}}$$

Accordingly, we have the following definition in terms of quantile functions.

Definition 4.14.

We say that X is DMRLHA (IMRLHA) if and only if

$$\displaystyle{\frac{\int _{0}^{u} \frac{q(p)} {M(p)}dp} {\int _{0}^{u}q(p)dp} }$$

is decreasing (increasing) in u. It is easy to see that

• (i) DMRL ⇒ DMRLHA ⇒ NBUE;

• (ii) Since $H_{Z}(u) = \frac{1} {M_{X}(u)}$, we have

  $$\displaystyle{\frac{\int _{0}^{u} \frac{q(p)} {M(p)}dp} {\int _{0}^{u}q(p)dp} = \frac{\int _{0}^{u}H_{Z}(p)q(p)dp} {\int _{0}^{u}q(p)dp}.}$$

  Thus, X is DMRLHA ⇔ Z is IFRA.

Honfeng and Yi [275] compared the failure rates of X and Z in defining what is called the new better (worse) than equilibrium hazard rate, NBEHR (NWEHR), if and only if $h_{X}(x) \leq h_{Z}(x)$. It is obvious that when E(X) < ∞,

$$\displaystyle\begin{array}{rcl} \text{NBEHR } \Leftrightarrow h_{X}(x) \leq h_{Z}(x)& \Leftrightarrow & h_{X}(x) \leq \frac{1} {m_{X}(x)} {}\\ & \Leftrightarrow & h_{X}m_{X}(x) \leq 1 {}\\ & \Leftrightarrow & 1 + m^{\prime}(x) \leq 1 {}\\ & \Leftrightarrow & X\text{ is DMRL}. {}\\ \end{array}$$

Like the IHRA notion comparison with the class of DMTTF distributions (see Definition 4.9), we see that DMRL ⇏ DMTTF and DMTTF ⇏ DMRL (Li and Xu [393]).

4.4.2 Used Better Than Aged Class

When a unit is working with unknown age, to assess its ageing behaviour, Alzaid [36] introduced the used better than aged (UBA) and its dual used worse than aged (UWA) classes of life distributions. Two induced classes from these are the UBAE (used better than aged in expectation) and UWAE (used worse than aged in expectation). When E(X) < ∞ and 0 < m(∞) < ∞, the UBA (UWA) class is specified by

$$\displaystyle{\bar{F}(x + t) \geq (\leq )\bar{F}(t){e}^{- \frac{x} {m(\infty )} },\;x \geq 0,\;t \geq 0.}$$

Accordingly, we have the following definitions.

Definition 4.15.

The random variable X is UBA (UWA) if and only if

$$\displaystyle{ Q(v + (1 - v)u) - Q(v) \geq (\leq ) - M(1)\log (1 - u) }$$

(4.21)

for all 0 ≤ u, v < 1, provided 0 < M(1) < ∞.

Definition 4.16.

We say that X is UBAE (UWAE) if and only if 0 < M(1) < ∞ and

$$\displaystyle{ M(u) \geq (\leq )M(1)\quad \mbox{ for all }u. }$$

(4.22)

The following properties hold for these four classes of life distributions:

• 1. IHR ⇒ DMRL ⇒ UBA ⇒ UBAE (see Alzaid [36] and Willmot and Cai [581] for proofs).

• 2. X is UBAE (UWAE) $\Leftrightarrow \bar{ F}_{Z}(x + t) \geq (\leq )\bar{F}_{Z}(t){e}^{- \frac{x} {m(\infty )} }$.

• 3. X is UBAE (UWAE) if and only if Z is UBA (UWA).

• 4. If

  $$\displaystyle{\bar{G}(x) = \frac{\int _{0}^{\infty }{e}^{-\alpha t}\bar{A}(x + t)dt} {\int _{0}^{\infty }{e}^{-\alpha t}\bar{A}(t)dt},\quad x \geq 0,\;-\infty <\alpha < \infty,}$$

  then $1 -\bar{ A}$ is UBA (UBAE) ⇒ G is UBA (UBAE). It may be observed that $\bar{G}(x)$ given above is a generalization of the equilibrium distribution.

• 5. If X is such that $E({X}^{r+s+2}) < \infty$, then

  $$\displaystyle{ \frac{\mu ^{\prime}_{r+s+2}} {(r + s + 2)!} \geq \frac{\mu ^{\prime}_{r+1}{(m(\infty ))}^{s+1}} {(r + 1)!} \quad \mbox{ if $X$ is UBA}.}$$

• 6. The moment generating function ϕ(t) of X satisfies

  $$\displaystyle\begin{array}{rcl} \phi (t)& \leq & 1 + \frac{\mu t} {1 - tm(\infty )}\quad \mbox{ if $X$ is UBA}, {}\\ \phi (t)& \leq & 1 + \frac{\mu t + {t}^{2}(\frac{\mu _{2}^{\prime}} {2} -\mu m(\infty ))} {1 - tm(\infty )} \quad \mbox{ if $X$ is UBAE}. {}\\ \end{array}$$

• 7. When X is UBA (UBAE) and E(X) < ∞ (E(X ²) < ∞), then all moments of X exist and are finite.

• 8. When X is UBAE, we have

  $$\displaystyle{ \frac{\mu _{r+s+3}^{\prime}} {(r + s + 3)!} \geq \frac{\mu _{r+2}^{\prime}} {(r + 2)!}{(m(\infty ))}^{s+1}.}$$

Properties 5–8 are taken from Ahmad [22] while Properties 1–4 are from Willmot and Lin [583].

A weaker condition than UBAE is given in Kotlyar [353] as NUABE defined by

$$\displaystyle{\int _{x}^{\infty }\bar{F}(t)dt \geq \mu {e}^{- \frac{x} {m(\infty )} }.}$$

Nair and Sankaran [445] introduced another version of mean residual life function which is the expected value of the asymptotic conditional distribution of residual life given age in a renewal process. This function

$$\displaystyle{ {m}^{{\ast}}(x) = \frac{\int _{x}^{\infty }(t - x)\bar{F}(t)dt} {\int _{x}^{\infty }\bar{F}(t)dt} = \frac{E({(X - x)}^{2}\vert X > x)} {2m(x)} }$$

(4.23)

is called the renewal mean residual life function (RMRL). In the quantile formulation, (4.23) is equivalent to

$$\displaystyle{ e(u) = {m}^{{\ast}}(Q(v)) = \frac{\int _{u}^{1}[Q(p) - Q(u)](1 - p)q(p)dp} {\int _{u}^{1}(1 - p)q(p)dp}. }$$

(4.24)

They then showed that m ^∗(x) is similar to the conventional mean residual life function m(x) and can be employed in all applications just as m(x). Differentiating

$$\displaystyle{e(u)\int _{u}^{1}(1 - p)q(p)dp =\int _{ u}^{1}[Q(p) - Q(u)](1 - p)q(p)dp}$$

and simplifying using

$$\displaystyle{M(u) = \frac{1} {1 - u}\int _{u}^{1}(1 - p)q(p)dp,}$$

we get

$$\displaystyle{M(u) = \frac{e(u)q(u)} {q(u) + e^{\prime}(u)}}$$

or

$$\displaystyle{ e^{\prime}(u) = q(u)\left [\frac{e(u) - M(u)} {M(u)} \right ]. }$$

(4.25)

Definition 4.17.

The random variable X belongs to the decreasing renewal mean residual life (DRMRL) or increasing renewal mean residual life (IRMRL) class according to e(u) being decreasing (increasing) in u.

The DRMRL (IRMRL) class has the following properties:

• 1. X is DRMRL (IRMRL) if and only if e(u) ≤ ( ≥ ) M(u);

• 2. If X is DMRL (IMRL), then X is DRMRL (IRMRL);

• 3. If X is DRMRL (IRMRL), then X is DVRL (IVRL);

• 4. The exponential distribution is characterized by a constant e(u);

• 5. X is DRMRL (IRMRL) if and only if C ^∗(u) ≤ ( ≥ )1;

• 6. X is DRMRL (IRMRL) if and only if M _z (u) = M _X (u);

• 7. If X _t and Z _t are residual lives of X and Z, respectively, then

  $$\displaystyle{M_{X_{t}}(u) \geq M_{z_{t}}(u) \Leftrightarrow X\text{ is DRMRL.}}$$

The closure properties with respect to various reliability operations in this case seems to be an open problem.

4.4.3 Decreasing Variance Residual Life

Recall from (2.10) and (2.42) that the variance residual life is given by

$${\displaystyle{\sigma }^{2}(x) = \frac{2} {\bar{F}(x)}\int _{x}^{\infty }\int _{ u}^{\infty }\bar{F}(t)dtdu - {m}^{2}(x),}$$

and the corresponding quantile definition is

$$\displaystyle\begin{array}{rcl} V (u)& =& {(1 - u)}^{-1}\int _{ u}^{1}{Q}^{2}(p)dp - {(M(u) + Q(u))}^{2} {}\\ & =& {(1 - u)}^{-1}\int _{ u}^{1}{M}^{2}(p)dp. {}\\ \end{array}$$

When σ ²(x) is decreasing (increasing), we say that X is decreasing variance residual life—DVRL (increasing variance residual life—IVRL).

Definition 4.18.

A lifetime random variable X is DVRL (IVRL) if V (u) is decreasing (increasing) in u.

From a practical point of view, the class of DVRL distributions is studied as it indicates positive ageing, and also the uncertainty in the system behaviour decreases with age. Some characteristics of these two ageing criteria are as follows:

• 1. V (u) is increasing (decreasing) if and only if C ^∗ 2(u) ≥ ( ≤ 1);

• 2. DMRL ⇒ DVRL.

  To prove this implication in terms of quantile functions, we note that

  $$\displaystyle\begin{array}{rcl} V (u)& =& \frac{1} {1 - u}\int _{u}^{1}{[Q(p) - Q(u)]}^{2}dp - {M}^{2}(u) {}\\ & =& \frac{2} {1 - u}\int _{u}^{1}(1 - p)(Q(p) - Q(u))q(p)dp - {M}^{2}(u) {}\\ & =& \frac{2} {1 - u}\int _{u}^{1}\int _{ p}^{1}q(p)(1 - t)q(t)dpdt - {M}^{2}(u) {}\\ & =& \frac{2} {1 - u}\int _{u}^{1}(1 - p)q(p)M(p)dp - {M}^{2}(u). {}\\ \end{array}$$

  Hence,

  $$\displaystyle{V (u) - {M}^{2}(u) = \frac{2} {1 - u}\int _{u}^{1}(1 - p)(M(p) - M(u))q(p)dp.}$$

  Since X is DMRL, M(p) − M(u) ≤ 0 for all u ≤ p, and so we have

  $$\displaystyle{V (u) - {M}^{2}(u) \leq 1\text{ or }{C}^{{\ast}2}(u) \leq 1,}$$

  and thus X is DVRL.

• 3. When F is strictly increasing, Z _n is DVRL if and only if Z is DMRL. This follows from the fact that

  $$\displaystyle{m_{Z}(t) = \frac{E(X_{t}^{2})} {E(X_{t})},}$$

  which is a decreasing function of t.

• 4. We have

  $$\displaystyle{ \frac{M_{Z}(u)} {M_{X}(u)} = \frac{1} {2}(1 + {C}^{{\ast}2}(u)),}$$

  so that if F is strictly increasing, then X is DVRL if and only if M _Z (u) ≤ M(u).

• 5. In general, the DVRL class is not closed under mixing.

• 6. A family of distributions F _θ obeys the non-crossing property if, for any α ₁ and α ₂, the graphs of $F_{\alpha _{1}}$ and $F_{\alpha _{2}}$ do not intersect on their common support. Stoyanov and Al-sadi [548] proved that if F _α is IVRL for each α > 0 and obeys the non-crossing property, then their mixture is IVRL.

• 7. Both DVRL and IVRL distributions are not closed under convolution.

• 8. Both DVRL and IVRL do not preserve the formation of coherent systems.

• 9. If X ₁ and X ₂ are independent copies of X and Y = min(X ₁, X ₂), then Al-Zahrani and Stoyanov [32] established that

  $$\displaystyle{\mu _{1}^{\prime}\mu _{2}^{\prime} \leq 4E(X_{2}{Y }^{2}) -\frac{8} {3}E({Y }^{3})}$$

  and

  $$\displaystyle{{\mu _{2}^{\prime}}^{2} \leq \frac{16} {3} E(X_{2}{Y }^{3}) - 4E({Y }^{4})}$$

  with equality sign holding in the two cases if and only if X is exponential.

• 10. We have

  $$\displaystyle{\bar{F}(x) \leq \frac{\mu } {\sigma (x) + x},\quad x \geq \mu -\sigma (x)}$$

  for DVRL distributions, and

  $$\displaystyle{\bar{F}(x) \geq \frac{\mu -x} {\sigma (x)}}$$

  for IVRL distributions (Launer [377]).

• 11. If X is DVRL (IVRL), then

  $$\displaystyle{E({X}^{2}\vert X \geq x) \leq (\geq ){E}^{2}(X\vert X \geq x) + {[E(X\vert X > x) - x]}^{2}.}$$

For a comprehensive account of the above results and other properties of monotone variance residual life classes, we refer to Gupta [234], Gupta et al. [246], Abouammoh et al. [8] and Gupta and Kirmani [244].

Just as in the case of the mean residual quantile function, other concepts can be formulated for the variance as well. From Launer [377] and Abouammoh et al. [8], we have the following new classes.

Definition 4.19.

A lifetime random variable X is net decreasing in variance residual quantile function (NDVRL) or net increasing in variance residual quantile function (NIVRL) according to V (u) ≤ ( ≥ ) σ ².

Definition 4.20.

We say that X has decreasing (increasing) variance residual life average DVRLA (IVRLA) if and only if

$$\displaystyle{\frac{\int _{0}^{u}V (p)q(p)dp} {\int _{0}^{u}q(p)dp} }$$

is decreasing (increasing). Further, X is new worse (better) than average variance residual life if and only if

$$\displaystyle{\frac{\int _{0}^{u}V (p)q(p)dp} {\int _{0}^{u}q(p)dp} \leq {(\geq )\ \sigma }^{2}.}$$

We have the implications DVRL ⇒ NDVRL and NBUE ⇒ NDVRL (see Definition 4.29 for NBUE).

4.4.4 Decreasing Percentile Residual Life Functions

Haines and Singpurwalla [258] discussed classes of life distributions based on the monotonic behaviour of the percentile residual life function p _α (x) defined in (2.19).

Definition 4.21.

The random variable X has decreasing (increasing) percentile life DPRL (α) (IPRL (α)) if F(0) = 0 and p _α (x) is decreasing (increasing) in x. In terms of quantile functions, the conditions become Q(0) = 0 or P _α (u) defined in (2.19) is decreasing (increasing) in u. If X has a constant p _α (x) (P _α (u)), it is both DPRL (α) and IPRL (α). Joe and Proschan [301] studied the distinguishing features of these two classes of distributions.

Recalling from (2.49) that

$$\displaystyle{P_{\alpha }(u) = Q(\alpha +(1-\alpha )u) - Q(u),}$$

we have

$$\displaystyle{P_{\alpha }^{\prime}(u) = [(1-\alpha )q(\alpha +(1-\alpha )u) - q(u)]q(u).}$$

Hence,

$$\displaystyle\begin{array}{rcl} X\text{ is DPRL}(\alpha )& \Leftrightarrow & (1-\alpha )q(\alpha +(1-\alpha )u) \leq q(u) {}\\ & \Leftrightarrow & \frac{1} {q(\alpha +(1-\alpha )u)} \geq \frac{1-\alpha } {q(u)} {}\\ & \Leftrightarrow & \frac{1 -\alpha -u +\alpha u} {1 - (\alpha +u -\alpha u)q(\alpha +u -\alpha u)} \geq \frac{(1-\alpha )(1 - u)} {(1 - u)q(u)} {}\\ & \Leftrightarrow & H(u +\alpha (1 - u)) \geq H(u) {}\\ & \Leftrightarrow & X\text{ is IHR.} {}\\ \end{array}$$

We see that X is IHR when X is DPRL (α) for all α in (0, 1). However, if X is DPRL (α) for some α, it is not necessary that X is IHR. Moreover, it is not necessary that DPRL (α) implies DPRL (β) for β > α. Haines and Singpurwalla [258] showed that DPRL (α) class is not closed under formation of coherent systems or convolution or mixture of distributions.

A particular case of the α-percentile residual life is the median percentile life when $\alpha = \frac{1} {2}$. Lillo [399] pointed out that this measure is preferred over the mean residual life in view of its robustness and use in regression models; see also Kottas and Gelfand [354] and Csorgo and Csorgo [160]. Denoting decreasing median residual life function by DMERL and increasing median residual life by IMERL, we have the following properties:

• 1. IHR implies DMERL;

• 2. DMRL does not imply DMERL and DMERL does not imply DMRL;

• 3. IMRL and IMERL also have no mutual implications either.

4.5 Concepts Based on Survival Functions

There are several criteria available in the literature based on the comparison of survival functions (quantile functions) or their integral versions, and we describe them in this section.

4.5.1 New Better Than Used

A distribution with F(0) = 0 is said to be new better than used (NBU) if

$$\displaystyle{ \bar{F}(x + t) \leq \bar{ F}(x)\bar{F}(t) }$$

(4.26)

for all x, t > 0. When the inequality in (4.26) is reversed, we say that the distribution is new worse than used (NWU). Here, we are comparing the residual life X _t and X of a unit and the definition says that a new unit has stochastically larger (smaller) life than one of age t, and therefore NBU (NWU) represents positive (negative) ageing. The equality in (4.26) holds if and only if

$$\displaystyle{\bar{F}(x + t) =\bar{ F}(x)\bar{F}(t)}$$

which is a Cauchy functional equation with the only continuous solution of the form $\bar{F}(x) = {e}^{-\lambda x}$, or X is exponential. One may refer to Rao and Shanbhag [506] for a thorough discussion on characterizations of distributions based on such functional equations.

Definition 4.22.

A random variable X with Q(0) = 0 is said to be NBU (NWU) if and only if

$$\displaystyle{ Q(u + v - uv) - Q(v) \leq Q(u) }$$

(4.27)

for 0 ≤ u < v < 1.

The equality in (4.27) holds when

$$\displaystyle{Q(u + v - uv) = Q(u) + Q(v)}$$

or

$$\displaystyle{Q(1 - (1 - u)(1 - v)) = Q(1 - (1 - u)) + Q(1 - (1 - v)),}$$

which reduces to the form

$$\displaystyle{Q(1 - x_{1}y_{1}) = Q(1 - x_{1}) + Q(1 - y_{1}).}$$

The last equation has the continuous solution as

$$\displaystyle{Q(u) = -k\log (1 - u),}$$

which means X is exponential.

The NBU property has several other implications as listed below:

• 1. IHRA (DHRA) ⇒ NBU (NWU).To see this result in the quantile form, we take 0 < u < v < 1 and express

  $$\displaystyle\begin{array}{rcl} X\text{ is IHRA }& \Rightarrow & \frac{\int _{0}^{u}H(p)q(p)dp} {Q(u)} \mbox{ is increasing in $u$} {}\\ & \Rightarrow & \frac{\int _{0}^{u+v(1-u)}H(p)q(p)dp} {Q(u + v(1 - u))} \geq \frac{\int _{0}^{u}H(p)q(p)dp} {Q(u)} {}\\ & \Rightarrow & \frac{Q(u + v - uv)} {Q(u)} \leq \frac{\int _{0}^{u+v(1-u)}{(1 - p)}^{-1}dp} {\int _{0}^{u}{(1 - p)}^{-1}dp} = 1 + \frac{\log (1 - u)} {\log (1 - v)} {}\\ & \Rightarrow & \frac{Q(u + v - uv)} {Q(u)} - 1 \leq \frac{\int _{0}^{u}H(p)q(p)dp} {\int _{0}^{v}H(p)q(p)dp} {}\\ & \leq & \frac{Q(v)} {Q(u)}\left [\frac{\int _{0}^{u}H(p)q(p)dp} {Q(u)} \big/\frac{\int _{0}^{v}H(p)q(p)dp} {Q(v)} \right ] {}\\ & \leq & \frac{Q(v)} {Q(u)}. {}\\ \end{array}$$

  Thus,

  $$\displaystyle{Q(u + v - u) \leq Q(u) + Q(v) \Rightarrow X\text{ is NBU.}}$$

• 2. An equivalent condition for X to be NBU is

  $$\displaystyle{\int g(\alpha x)h[(1-\alpha )x]dF(x) \leq \int g(x)dF(x)\int h(x)dF(x)}$$

  for all non-negative increasing functions g and h and all 0 < α < 1 (Block et al. [111]).

• 3. If X is NBU, it has finite moments of all positive orders, which is a stronger result than those for IHR and IHRA classes.

• 4. If X has a density, the NBU (NWU) implies NBUHR (NWUHR):

  $$\displaystyle\begin{array}{rcl} \mbox{ $X$ is NBU}& \Leftrightarrow & Q(u + v - uv) \leq Q(u) + Q(v) {}\\ & \Leftrightarrow & \frac{Q(u + v - uv) - Q(u)} {v(1 - u)} \leq \frac{Q(v)} {v(1 - u)} {}\\ & \Rightarrow & Q^{\prime}(u) \leq \frac{Q^{\prime}(0)} {1 - u}\mbox{ on taking limits as $v \rightarrow 0$} {}\\ & \Rightarrow & (1 - u)q(u) \leq q(0) \Rightarrow H(u) \geq H(0) {}\\ & \Leftrightarrow &\mbox{ $X$ is NBUHR.} {}\\ \end{array}$$

• 5. If each component of a coherent system is NBU, then the system life is also NBU.

• 6. Convolution of two NBU distributions is NBU.

• 7. The residual life of an NBU distribution is not NBU. A necessary and sufficient condition for this to hold is that X is IHR.

• 8. The mixture of two NBU distributions is NBU, provided that the distributions of the components do not cross.

• 9. For a sequence (X _n ) of independent lifetime random variables with NBU distributions, $S_{N} = X_{1} + X_{2} + \cdots + X_{N}$, where N is a positive integer-valued random variable, is also NBU.

• 10. When X is NBU, we have

  $$\displaystyle\begin{array}{rcl} \bar{F}(x)& \geq & {[\bar{F}(t)]}^{\frac{1} {k} },\quad \frac{t} {k + 1} < x < \frac{t} {k},\ k = 0,1,2,\ldots {}\\ & \leq & {[\bar{F}(t)]}^{k},\quad kt < x < (k + 1)t, {}\\ \end{array}$$

  and when X is NWU,

  $$\displaystyle\begin{array}{rcl} \bar{F}(x)& \leq & {[\bar{F}(t)]}^{ \frac{1} {k+1} },\quad \frac{t} {k + 1} < x < \frac{t} {k}, {}\\ & \geq & {[\bar{F}(t)]}^{k+1},\quad kt < x < (k + 1)t. {}\\ \end{array}$$

• 11. When X is NBU (NWU), we have

  $$\displaystyle{ \frac{\mu _{r+s+2}^{\prime}} {\varGamma (r + s + 3)} \geq (\leq ) \frac{\mu _{r+1}^{\prime}} {\varGamma (r + 2)} \frac{\mu _{s}^{\prime}} {\varGamma (s + 2)}\quad r,s \geq 0.}$$

• 12. NBU does not imply DMTTF.

• 13. If X is NBU and C ^∗(u) is increasing, then Z is NBU.

• 14. If a coherent system from independent NBU components has exponential life, then it is essentially a series system with exponential components. We refer the readers to Shaked [530], Abouammoh and El-Neweihi [4] and Barlow and Proschan [70] for some further details in this regard.

There are several variants of the NBU concept presented in the literature. In situations wherein a unit or system deteriorates over time, say, up to an instant t ₀, to make the system more effective, replacement or repairs are often thought of. But, by this operation, the system may not revert to the same effectiveness as at t ₀. An ageing concept that is relevant in such a situation is to assume that the system lifelength is smaller from t ₀ onwards compared to a new one. This idea gives rise to the NBU-t ₀ (NWU-t ₀) class of life distributions that satisfy (Hollander et al. [274])

$$\displaystyle{\bar{F}(t_{0} + x) \leq (\geq )\bar{F}(t_{0})\bar{F}(x),\quad x \geq 0.}$$

Definition 4.23.

We say that X is NBU-u ₀ (NWU-u ₀) if and only if, for some 0 ≤ u ₀ < 1, we have

$$\displaystyle{Q(u + u_{0} - uu_{0}) \leq (\geq )\ Q(u) + Q(u_{0})\quad \text{ for all }0 \leq u < 1.}$$

The class of distributions that are both NBU-t ₀ and NWU-t ₀ is not confined to exponential distributions as in the case of other ageing notions. Along with exponential laws, all distributions with periodic hazard quantile functions and those distributions whose quantile functions are specified by Q(u) = Q ₁(u), 0 ≤ u ≤ u ₀, where Q ₁(0) = 0, are also both NBU-t ₀ and NWU-t ₀. Some other important properties of these two classes are as follows:

• 1. If $H_{X}(u) \leq H_{Y }(u)$, 0 ≤ u ≤ v ₀, $H_{X}(u) = H_{Y }(u)$ in (v ₀, 1), and H _X (u) is decreasing in [0, v ₁], 0 < v ₁ < v ₀, then X is NBU-v ₀, but not NBU. In general, NBU ⇒ NBU-u ₀;

• 2. NBU-t ₀ property is preserved under the formation of coherent systems, but NWU-t ₀ is not;

• 3. Both NBU-t ₀ and NWU-t ₀ are not preserved under convolution;

• 4. NBU-t ₀ is preserved under mixtures of non-crossing distributions, but not for arbitrary mixtures. NWU-t ₀ is not closed with respect to formation of mixtures; see Park [483] for more details.

Kayid [318] has presented a generalization of the NBU and NBU-t ₀ classes. If A denotes the set of functions a(u) satisfying a(u) > 0 in (0, 1) and a(u) = 0 otherwise, X is said to be NBU with respect to a(u), denoted by NBU_(a), if and only if

$$\displaystyle{\int _{0}^{F_{t}^{-1}(u) }a(F_{t}(x))dx \leq \int _{0}^{{F}^{-1}(p) }a(F(x))dx,}$$

where F _t is the usual residual life distribution of $X_{t} = X - t\vert (X > t)$. When a( ⋅) is a constant, it is evident that NBU_(a) reduces to NBU, and when the time is fixed as t ₀, NBU_(a) ⇔ NBU-t ₀. For a non-negative X with continuous F, if X is NBU (NBU_(a)) and a(u) is decreasing, then X is NBU_(a) (NBU).

A slightly different concept is NBU* t ₀ (NWU* t ₀), defined by Li and Li [398] through the relationship

$$\displaystyle{\bar{F}(x + y) \leq (\geq )\bar{F}(x)\bar{F}(y)}$$

for all x ≥ 0, y ≥ t ₀ > 0. The difference between NBU-t ₀ and NBU*t ₀ is that in the former t ₀ is a fixed time while in the latter it extends beyond t ₀. From the above, we have the following definition.

Definition 4.24.

We say that X is NBU*u ₀ (NWU*u ₀) if and only if the quantile function satisfies

$$\displaystyle{Q(u + v - uv) \leq (\geq )Q(u) + Q(v)\quad \mbox{ for all $0 \leq u < 1$ and $v \geq u_{0}$.}}$$

The two classes NBU*u ₀ and NWU*u ₀ possess the following properties:

• 1. NBU ⇒ NBU * u ₀ ⇒ NBU * u ₁, u ₁ ≥ u ₀;

• 2. IHRA * u ₀ need not imply NBU * u ₀;

• 3. If $X_{1},X_{2},\ldots,X_{n}$ are independent and NBU * t _i , $i = 1,2,\ldots,n$, then the life of the coherent system with X _i as lifetimes of the components is also NBU * t ₀, where $t_{0} =\max (t_{1},t_{2},\ldots,t_{n})$. As a consequence, a coherent system with n independent components each of which is NBU * t ₀ is also NBU * t ₀;

• 4. If X ₁ and X ₂ are independent NBU * t ₀ lifetimes, their convolution is also NBU * t ₀;

• 5. If a life distribution is NBU * t ₀, it is also NBU-t ₀. Thus, we have

  $$\displaystyle{\text{NBU} \Rightarrow \mbox{ NBU} {\ast} t_{0} \Rightarrow \text{NBU -}t_{0}.}$$

Another generalization of the NBU class has been provided by Deshpande et al. [172] using second order stochastic dominance, called the new better (worse) than used in second order dominance, NBU(2) (NWU(2)).

Definition 4.25.

A lifetime random variable X is said to be NBU(2) (NWU(2)) if and only if

$$\displaystyle{\int _{0}^{x}\bar{F}(y)dy \geq \int _{ 0}^{x}\frac{\bar{F}(t + y)} {\bar{F}(t)} dy}$$

for all t, x ≥ 0, or equivalently,

$$\displaystyle{\int _{0}^{u}(1 - p)q(p)dp \geq \frac{1} {1 - v}\int _{0}^{u}[1 - {Q}^{-1}(Q(p) + Q(v))]q(p)dp}$$

for all 0 ≤ u, v < 1.

Obviously,

$$\displaystyle{\text{NBU (NWU)} \Rightarrow \text{NBU}\;(2)(\text{NWU}(2)).}$$

Li and Kochar [389] have shown that NBU(2) class is closed under the formation of series systems and convolution. Li [396, 397] further established that NBU(2) class is closed with respect to formation of mixtures and parallel systems. The convolution of X ₁ and X ₂ which are NBU(2) is also NBU(2) (Hu and Xie [287]). Some limited converse results on the closure properties have been discussed in Li and Yam [394]. If a system possesses a particular ageing property, the problem is to examine whether components satisfy the same property. Li and Yam [394] have shown that if parallel and series systems consisting of independent and identically distributed components are NWU (2), then the components are also NWU (2).

A stochastic version of the NBU property has been discussed by Singh and Deshpande [542] along the lines of stochastically increasing hazard rates presented earlier.

Definition 4.26.

A lifetime X is said to be stochastically new better than used (SNBU) if

$$\displaystyle{P\left (X \geq \sum _{i=0}^{k+1}Y _{ i}\vert X \geq \sum _{i=0}^{k}Y _{ i}\right ) \leq P(X \geq Y _{k+1}),}$$

where $Y _{0},Y _{1},\ldots,Y _{n},\ldots$, with Y ₀ = 0, is a sequence of independent and identically distributed exponential random variables each with mean μ, and X is independent of the Y _i ’s. It has been established that

$$\displaystyle{\text{SIHR } \Rightarrow \text{ SNBU}\quad \mbox{ and }\quad \text{NBU } \Rightarrow \text{ SNBU.}}$$

Yet another extension of the NBU and NWU property of ageing systems in the context of comparison of the reliability of new and used systems by the use of dynamic signatures has been provided by Samaniego et al. [514]. Recall that the signature of a system with n independent and identically distributed components is an n-dimensional vector whose ith component is the probability that the ith ordered component failure is fatal to the system. System signatures have found key applications in the study and comparison of engineered systems; see, for example, Samaniego [513]. Now, when a working used system is inspected at time t and it is observed that precisely k failures have occurred by that time, then the (n − k)-dimensional vector whose jth element is the probability that the (k + j)th ordered component failure is fatal to the system has been termed the dynamic signature by Samaniego et al. [514]. It is indeed a distribution-free measure of the design of the residual system. With such a notion of dynamic signature, these authors presented the following dynamic versions of NBU (NWU).

Definition 4.27.

Let T denote the lifetime of a coherent system with n components whose lifetimes $X_{1},\ldots,X_{n}$ are independent and identically distributed with a continuous distribution function F over (0, ∞). Let $X_{1:n} \leq \cdots \leq X_{n:n}$ denote the order statistics of $X_{1},\ldots,X_{n}$, and E _i be the event that $\{X_{1:n} \leq t < X_{i+1:n}\}$, with X _0: n  ≡ 0. Then, for fixed $i \in \{ 0,1,\ldots,n - 1\}$, T is said to be conditionally NBU, given i failed components, denoted by i-NBU, if for all t > 0, either $P(E_{i} \cap \{ T > t\}) = 0$, or

$$\displaystyle{P(T > x) \geq P(T > x + t\mid E_{i} \cap \{ T > t\})\quad \mbox{ for all }x > 0.}$$

Definition 4.28.

A n-component is said to be uniformly new better than used, denoted by UNBU, if it is i-NBU for $i \in \{ 0,1,\ldots,n - 1\}$.

Samaniego et al. [514] have illustrated the use of these concepts in the performance evaluation of burn-in systems.

4.5.2 New Better Than Used in Convex Order

Cao and Wang [127] discussed a new class of distributions called new better than used in convex order (NBUC) and its dual new worse than used in convex order (NWUC). The NBUC (NWUC) class satisfies

$$\displaystyle{\int _{x}^{\infty }\bar{F}_{ y}(t)dy \leq (\geq )\int _{x}^{\infty }\bar{F}(t)dt.}$$

In terms of quantiles, we have the following definition.

Definition 4.29.

We say that X is NBUC (NWUC) if and only if

$$\displaystyle{ \frac{1} {1 - v}\int _{u}^{1}[1 - {Q}^{-1}(Q(p) + Q(v))]q(p)dp\ \leq (\geq )\int _{ u}^{1}(1 - p)q(p)dp.}$$

These two classes possess the following properties:

• 1. NBU (NWU) ⇒ NBUC (NWUC), as NBUC is the integrated version of NBU;

• 2. A parallel system of independent and identically distributed NBUC components is NBUC (Hendi et al. [269] and Li et al. [390]). Even when the components are independent and non-identical, NBUC class is preserved under the formation of parallel systems (Cai and Wu [124]);

• 3. The convolution of two independent NBUC variables is NBUC (Hu and Xie [287]);

• 4. The NBUC property is preserved under monotonic antistar-shaped transformation and under nonhomogeneous Poisson shock models (Li and Qiu [391]);

• 5. Under the formation of mixtures, the NBUC class is preserved (Li [397]);

• 6. If X is NBUC, then (Ahmad and Mugadi [26])

  $$\displaystyle{(r + 2)!(s + 1)!E({X}^{r+s+3}) \leq (r + s + 3)!E({X}^{r+2})E({X}^{s+1}).}$$

As an application of the concept, Belzunce et al. [87] compared the age replacement (block) policies and a renewal process with no planned replacements when the lifetime of the unit is NBUC.

A further extension of the NBUC class is the NBUCA class defined by

$$\displaystyle{\int _{0}^{\infty }\int _{ x}^{\infty }\bar{F}(u + t)dudx \leq \bar{ F}(t)\int _{ 0}^{\infty }\int _{ x}^{\infty }\bar{F}(u)dudx\quad \mbox{ for all $t \geq 0$.}}$$

For properties and further details, we refer the readers to Ahmad and Mugadi [26].

Elabatal [186] studied the extensions of NBU(2) and NBUC classes at a specific age t ₀, called NBU(2)-t ₀ and NBUC-t ₀, which can be defined as follows.

Definition 4.30.

The NBU(2)-v ₀ class of distributions is one that satisfies

$$\displaystyle{ \frac{1} {1 - v_{0}}\int _{0}^{u}[1 - {Q}^{-1}(Q(p) + Q(v_{ 0}))]q(p)dp \leq \int _{0}^{u}(1 - p)q(p)dp}$$

for some 0 ≤ v ₀ < 1.

Definition 4.31.

X is said to be NBUC-v ₀ if, for some 0 ≤ v ₀ < 1, we have

$$\displaystyle{ \frac{1} {1 - v_{0}}\int _{u}^{1}[1 - {Q}^{-1}(Q(p) + Q(v_{ 0}))]q(p)dp \leq \int _{u}^{1}(1 - p)q(p)dp.}$$

It is known that if X ₁ and X ₂ are independent NBUC-t ₀ variables, then the convolution is also NBUC-t ₀. The class is also closed under the formation of a parallel system of iid components which are NBUC-t ₀. As Poisson shock model interpretation $\bar{H}(x)$ is NBUC-t ₀ if (P _k ) has discrete NBUC-t ₀ property that satisfies

$$\displaystyle{\sum _{j=k}^{\infty }\bar{P}_{ i+j} \leq \bar{ P}_{i}\sum _{j=k}^{\infty }\bar{P}_{ j},\quad \bar{P}_{K} = 1 - P_{K}.}$$

Based on survival function, Hendi [268] introduced the increasing cumulative (decreasing) survival class, denoted by ICSS (DCSS), through the property

$$\displaystyle{\int _{0}^{x}\bar{F}_{ t_{1}}(y)du \leq (\geq )\int _{0}^{x}\bar{F}_{ t_{2}}(y)dy}$$

for all x > 0, $0 \leq t_{1} \leq t_{2} < \infty$. It can be seen that the ICSS (DCSS) class is equivalent to the IHR(2) (DHR(2)). However, Hendi [268] proved that DCSS is preserved under convolution, while ICSS is not closed under the formation of convolution and coherent structures. These properties could be read in conjunction with those of the IHR(2) class discussed earlier.

Yet another variant of the NBU distributions is the new better (worse) than used in Laplace order, denoted by NBUL (NWUL).

Definition 4.32.

Yue and Cao [598] defined the NBUL (NWUL) class as one that satisfies the inequality

$$\displaystyle{\int _{0}^{\infty }{e}^{-sx}\bar{F}(t + x)dx \leq (\geq )\int _{ 0}^{\infty }{e}^{-sx}\bar{F}(x)dx.}$$

This concept has different interpretations in the context of ageing. One of them is by considering the mean life of a series system of two independent components, one having exponential survival function and the other having survival function $\bar{F}$. In two such systems A and B, if A has used component of age t while B has a used component with survival function $\bar{F}$, then F is NBUL means that the mean life of A is not larger than that of B. These classes have the following properties:

• 1. NBU ⇒ NBU(2) ⇒ NBUL;

• 2. Let X and Y be independent random variables with survival functions $\bar{F}$ and e ^{− λ x}, respectively, and W = min(X, Y ). Then, X is NBUL (NWUL) if and only if W is NBUE (NWUE), and some details of NBUE (NWUE) classes are presented in the next section;

• 3. NBUL is not closed under the formation of series systems. However, if the component survival functions are completely monotone, then the closure property holds. The NBUL concept is used in connection with replacement policies. For a detailed study of the properties of the classes and their applications, we refer to Yue and Cao [598], Al-Wasel [31], and Li and Qiu [391].

Joe and Proschan [301] have provided a classification of life distributions based on percentiles, which are as follows.

A lifetime random variable X is new better (worse) than used with respect to the α-percentile, denoted by NBUP-α (NWUP-α), if F(0) = 0 and $p_{\alpha }(0) \geq (\leq )p_{\alpha }(x)$ for all x ≥ 0.

They then established the following properties:

• 1. NBU ⇔ NBUP-α for all 0 < α < 1;

• 2. DPRL-α ⇒ NBUP-α for any 0 < α < 1;

• 3. If X is NBUP-α, then $\bar{F}(x) \leq {(1-\alpha )}^{n}$, $np_{\alpha }(0) \leq x < (n + 1)p_{\alpha }(0)$ for n = 0, 1, 2, …, and if F is continuous, $\bar{F}(x) \leq {(1-\alpha )}^{n+1}$, $np_{\alpha }(0) \leq x < (n + 1)p_{\alpha }(0)$;

• 4. An NBUP-α distribution has a finite mean that is bounded above by $\frac{p_{\alpha }(0)} {\alpha }$. If F is continuous, F has a mean (possibly infinite) that is bounded from below by $\frac{(1-\alpha )p_{\alpha }(0)} {\alpha }$;

• 5. An NBUP-α distribution has finite moment of order r > 0;

• 6. The closure properties with respect to formation of coherent systems, convolution and mixtures do not hold for NBUP-α and NWUP-α distributions.

4.5.3 New Better Than Used in Expectation

Instead of comparing a life distribution with its residual life distribution, a weaker concept results when expectations are considered for this comparison. This leads to new better than used in expectation (NBUE) and its dual new worse than used in expectation. If E(X) < ∞, X is said to be NBUE (NWUE) if and only if

$$\displaystyle{\mu \geq (\leq )\int _{0}^{\infty }\frac{\bar{F}(x + t)} {\bar{F}(t)} dx = m(x)}$$

for all t ≥ 0 for which $\bar{F}(t) > 0$. This says that a used unit of any age has a smaller mean residual life than a new unit with the same life distribution.

Definition 4.33.

We say that a lifetime X is NBUE if and only if

$$\displaystyle{ \frac{1} {1 - v}\int _{0}^{1}\{1 - {Q}^{-1}(Q(p) + Q(v))\}q(p)dp \leq \mu =\int _{ 0}^{1}(1 - p)q(p)dp,}$$

or

$$\displaystyle{ \frac{1} {1 - u}\int _{u}^{1}(1 - p)q(p)dp \leq \mu.}$$

The NBUE and NWUE classes have the following properties:

• 1. NBU (NWU) ⇒ NBUC (NWUC) ⇒ NBUE (NWUE);

• 2. NBU(2) (NWU(2)) ⇒ NBUE (NWUE);

• 3. DMRL (IMRL) ⇒ NBUE (NWUE);

• 4. NBUE (NWUE) ⇒ NDVRL (NIVRL);

• 5. NBUE (NWUE) ⇒ M(u) ≤ ( ≥ )M(0). The last inequality is equivalent to $H_{Z}(u)\ \geq (\leq )\ H_{X}(0)$, and so

  $$\displaystyle{X\text{ is NBUE (NWUE)} \Rightarrow Z\text{ is NBUHR (NWUHR)};}$$

• 6. Both NBUE and NWUE classes are not closed under the formation of coherent systems;

• 7. The convolution of two NBUE distributions is NBUE, but this preservation property is not true for NWUE;

• 8. The mixture of two NBUE (NWUE) life distributions is not in general NBUE (NWUE), while the mixture of NWUE distributions, no two of which cross, is again NWUE. This property is not shared by NBUE class. For proofs of Properties 6–8, see Marshall and Proschan [413];

• 9. If X is NBUE (NWUE), then $\int _{x}^{\infty }\bar{F}(t)dt \leq \mu {e}^{-\frac{x} {\mu } }$;

• 10. When X is NBUE, we have

  $$\displaystyle{\bar{F}(x) \geq \left \{\begin{array}{@{}l@{\quad }l@{}} 1 -\frac{x} {\mu } \quad &,\ x \leq \mu \\ 0 \quad &,\ x \geq \mu \end{array} \right.,}$$

  and when X is NWUE, we have

  $$\displaystyle{\bar{F}(x) \leq \frac{\mu } {\mu +x},\quad x \geq 0\quad \text{(Haines and Singpurwalla [258])};}$$

• 11. When X is NBUE, we have

  $$\displaystyle F(x) \geq \frac{{\sigma }^{2} +{\mu }^{2} - {x}^{2}} {{\sigma }^{2} + {(\mu +x)}^{2} - {x}^{2}},\quad x \leq {(\mu _{2}^{\prime})}^\frac{1} {2},$$

  and in the case of NWUE, we have

  $$\displaystyle\begin{array}{rcl} F(x)& \leq & \frac{{\sigma }^{2}} {{\sigma }^{2} + {(\mu +x)}^{2}},\quad 0 < x < \frac{{2\sigma }^{2}} {\mu } {}\\& \geq & \frac{{\sigma }^{2}} {{\sigma }^{2} + {x}^{2}},\quad x \geq \frac{{2\sigma }^{2}} {\mu } \quad \text{(Launer [377])}; {}\\ \end{array}$$

• 12. In the case of NBUE (NWUE) distributions, we have

  $$\displaystyle{ \frac{\mu _{r+1}^{\prime}} {\varGamma (r + 2)} \leq (\geq ) \frac{\mu _{r}^{\prime}} {\varGamma (r + 1)}\mu \quad \text{(Barlow and Proschan [70])};}$$

• 13. X is NBU does not imply Z is NBUE, nor X is NBUE implies Z is NBUE;

• 14. Neither DVRL ⇒ NBUE nor NBUE ⇒ DVRL. A common property shared by the two concepts is that the coefficient of variation of X is ≤ 1, provided F is strictly increasing;

• 15. If X is NBUE (NWUE), then $\bar{F}_{Z}(x) \leq (\geq )\bar{F}_{X}(x)$;

• 16. If X is NBUE and E(X) = E(Z), then X is exponential and converse is true as well. For details on Properties 14–16, see Gupta [233];

• 17. DMTTF ⇒ NBUE.

4.5.4 Harmonically New Better Than Used

The harmonically new better (worse) than used in expectation HNBUE (HNWUE) class of life distribution, introduced by Rolski [512], consists of distributions for which

$$\displaystyle{ \int _{x}^{\infty }\bar{F}(t)dt \leq (\geq )\ \mu {e}^{-\frac{x} {\mu } },\quad x \geq 0. }$$

(4.28)

An equivalent definition is presented below.

Definition 4.34.

A lifetime random variable X is HNBUE (HNWUE) if and only if one of the following conditions are satisfied:

• (i)
  $$\displaystyle{\int _{u}^{1}(1 - p)q(p)dp \leq (\geq )\ \mu {e}^{-\frac{Q(u)} {\mu } };}$$
• (ii)
  $$\displaystyle{{\left \{ \frac{\int _{0}^{u}\frac{q(p)dp} {M(p)} } {\int _{0}^{u}q(p)dp}\right \}}^{-1} \leq (\geq )\ \mu.}$$

The first definition follows directly from (4.28). To prove the equivalence of (i) and (ii), we observe that

$$\displaystyle\begin{array}{rcl} \text{(ii)}& \Leftrightarrow & \int _{0}^{u}q(p){\left ( \frac{1} {1 - p}\int _{p}^{1}(1 - t)q(t)dt\right )}^{-1}dp \geq \frac{Q(u)} {\mu } {}\\ & \Leftrightarrow & \int _{0}^{u}\left [ \frac{q(p)(1 - p)} {\int _{p}^{1}(1 - t)q(t)dt}\right ]dp \geq \frac{Q(u)} {\mu } {}\\ & \Leftrightarrow & \log \mu -\log \int _{0}^{u}(1 - p)q(p)dp) \geq \frac{Q(u)} {\mu } {}\\ & \Leftrightarrow & \mathrm{(i)}. {}\\ \end{array}$$

Thus, the HNBUE concept says that the harmonic mean of the mean residual hazard quantile function of a unit of age x is not grater than the harmonic mean life of a new unit. The two classes HNBUE and HNWUE enjoy the following properties:

• 1. NBUE (NWUE) ⇒ HNBUE (HNWUE), which follows from

  $$\displaystyle\begin{array}{rcl} \text{NBUE }& \Rightarrow & M(u) \leq M(0) {}\\ & \Rightarrow & \frac{q(u)} {M(u)} \geq \frac{q(u)} {M(0)} {}\\ & \Rightarrow & \frac{\int _{0}^{u} \frac{q(p)} {M(p)}dp} {Q(u)} \geq \frac{\int _{0}^{u}q(p)dp} {M(0)Q(u)} \Rightarrow \text{ HNBUE}; {}\\ \end{array}$$

• 2. A necessary and sufficient condition that X is HNBUE (HNWUE) is that

  $$\displaystyle{E\phi (X) \leq (\geq )\ E\phi ({X}^{{\ast}})}$$

  for all non-decreasing convex functions ϕ on (0, ∞) with $\phi (0+) = 0$, where X ^∗ is exponential with the same mean μ as X;

• 3. X is HNBUE if and only if $Q_{z}(u) \leq Q_{{X}^{{\ast}}}(u)$;

• 4. The HNBUE class is closed under the operation of forming non-negative linear combination of HNBUE random variables;

• 5. Both classes are not preserved under the formation of coherent structures;

• 6. The HNWUE class is preserved under mixing, but HNWUE is not;

• 7. We have

  $$\displaystyle{\bar{F}(x) \leq \left \{\begin{array}{@{}l@{\quad }l@{}} 1 \quad &,\ x <\mu \\ e\frac{\mu -x} {x},\quad &,\ x >\mu \end{array} \right.,}$$

  when X is HNBUE;

• 8. ${\mu }^{r+3} \geq \frac{\mu _{r+3}} {(r+3)!}$ if X is HNBUE;

• 9. $\bar{H}(t) =\sum _{ k=0}^{\infty }P(N(t) = k)$ $\bar{P}_{k}$ is HNBUE (HNWUE), where N(t) is a counting process governing the shocks and the interarrival times of shocks are independent HNBUE (HNWUE). For further details, one may refer to Klefsjö [332], Bhattacharjee and Kandar [98], Al-Ruzaize et al. [30], Basu and Bhattacharjee [79] and Cheng and Lam [144]. The kth order HNBUE has been studied by Basu and Ebrahimi [80].

Definition 4.35.

A lifetime random variable is (k-HNBUE) if

$$\displaystyle{\frac{1} {x}\int _{0}^{x}{m}^{-k}(t)dt \leq {(\geq )\mu }^{k}\quad \mbox{ for all $x > 0$},}$$

or

$$\displaystyle{\frac{\int _{0}^{u}{M}^{-k}(p)q(p)dp} {\int _{0}^{u}q(p)dp} \leq {(\geq )\mu }^{k}\quad \mbox{ for $0 < u < 1$},}$$

where k = 1 corresponds to the usual HNBUE. It is known that whenever X is (k + 1)-HNBUE, it is also k-HNBUE.

Using stochastic dominance of order three, the HNBUE concept can be generalized, which results in HNBUE(3) (HNWUE(3)) defined by

$$\displaystyle{\int _{0}^{\infty }\int _{ t}^{\infty }\bar{F}(u)dudt \leq {(\geq )\mu }^{2}{e}^{-\frac{x} {\mu } }\quad \mbox{ for all $x,t \geq 0$}.}$$

It is evident that HNBUE ⇒ HNBUE(3).

4.5.5 $\mathcal{L}$ and$\mathcal{M}$ Classes

A still larger class than the HNBUE can be constructed using transforms. Klefsjö [337] introduced the $\mathcal{L}$-class by considering the Laplace transform of the survival function.

Definition 4.36.

We say that a random variable X with finite mean μ belongs to the $\mathcal{L}$-class ($\bar{\mathcal{L}}$-class) if and only if

$$\displaystyle{\int _{0}^{\infty }{e}^{-sx}\bar{F}(x)dx \geq (\leq ) \frac{\mu } {1 + s\mu }.}$$

Chaudhury [138] found that for X in the $\mathcal{L}$-class, the coefficient of variation is ≤ 1. However, the exponential distribution is not characterized by the property that the coefficient of variation is unity. Further,

$$\displaystyle{X\text{ is HNBUE (HNWUE) } \Rightarrow X\text{ is }\mathcal{L}(\bar{\mathcal{L}}).}$$

Chaudhury [139] also established that if (F _n ) is a sequence of life distributions in $\mathcal{L}$ and F _n converges weakly to F, then F also belongs to $\mathcal{L}$. Consider a sequence $X_{1},X_{2},\ldots$, of independent and identically distributed random variables, and N as a geometric random variable over the set of positive integers. If N is independent of the X _i ’s, then the sum $S =\sum _{ i=1}^{N}X_{i}$ is called a geometric compound. Lin and Hu [403] established the preservation of the $\mathcal{L}$ class under geometric compounding. Several other interesting properties are presented in Bhattacharjee and Sengupta [97] Lin [400], and Nanda [457].

Klar [330] has given an example of a distribution that belongs to $\mathcal{L}$ with the property that its hazard rate tends to zero and mean residual life tends to infinity, which led to some doubts about the $\mathcal{L}$-class representing positive ageing. To overcome this limitation, Klar and Muller [331] presented an ageing class in which the Laplace transform is replaced by the moment generating function, and referred to it as the $\mathcal{M}$ class.

Definition 4.37.

We say that X belongs to the $\mathcal{M}$ class if

$$\displaystyle{\int _{0}^{\infty }{e}^{tx}dF(t) \leq \frac{1} {1 -\mu x},\quad 0 \leq x < \frac{1} {\mu },}$$

or

$$\displaystyle{\int _{0}^{\infty }\bar{F}(t){e}^{tx}dx \leq \frac{\mu } {1 -\mu x},}$$

where $\frac{1} {1-\mu x}$ is the moment generating function of the exponential distribution.

Notice that since E(e ^tX) has to be finite, $x < \frac{1} {\mu }$, and so the case h(x) → 0 as x → ∞ does not arise. We have the following properties for the $\mathcal{M}$ class:

• 1. The $\mathcal{M}$ class contains all HNBUE distributions.

• 2. Distributions in the $\mathcal{M}$ class are closed under convolution of independent random variables;

• 3. The $\mathcal{M}$ class contains all random variables with P(a < X < b) = 1 and $E(X) \geq \frac{a+b} {2}$, 0 ≤ a < b, and also all symmetric distributions;

• 4. Let $X_{Y } = X - Y \vert (X > Y )$. Then, X is in $\mathcal{L}$ if E(X _Y ) < μ for all Y independent of X, and have a density function of the form

  $$\displaystyle{g_{t}(x) = \frac{{e}^{tx}\bar{F}(x)} {\int {e}^{tx}\bar{F}(x)dx}.}$$

  On the other hand, X is in $\mathcal{M}$ if E(X _Y ) < Y for all Y independent of X, and have density g _t (x) above for some 0 < t < μ ^− 1;

• 5. X is in $\mathcal{M}\Rightarrow X$ is in $\mathcal{L}$, and

  $$\displaystyle{X_{i}\text{ is in }\mathcal{M}\Rightarrow \sum \alpha _{i}X_{i}\text{ is in }\mathcal{M},\quad \mbox{ with }\alpha _{i} \geq 0,\;\sum \alpha _{i} = 1,}$$

  and the X _i ’s have a common mean.

4.5.6 Renewal Ageing Notions

The renewal ageing concepts essentially compare the reliability functions of four random variables—X, its residual life X _t , the equilibrium random variable Z, and the corresponding residual life Z _t . Results in this direction are due to Abouammoh et al. [7], Bon and Illayk [116], Abouammoh and Qamber [10] and Abdel-Aziz [2].

Definition 4.38.

We say that X is new better (worse) than renewal used, denoted by NBRU (NWRU), if and only if, for all t ≥ 0,

$$\displaystyle{\int _{x+t}^{\infty }\frac{\bar{F}(u)du} {\bar{F}(x)} \leq (\geq )\int _{x}^{\infty }\bar{F}(u)du.}$$

Definition 4.39.

Renewal new is better (worse) than used, denoted by RNBU (RNWU), if and only if, for all t,

$$\displaystyle{\frac{\bar{F}(x + t)} {\bar{F}(t)} \leq (\geq )\ \frac{1} {\mu } \int _{x}^{\infty }\bar{F}(u)du.}$$

Definition 4.40.

Renewal new is better (worse) than used in expectation, denoted by RNBUE (RNWUE), if and only if

$$\displaystyle{2\mu \int _{x}^{\infty }\bar{F}(u)du \leq \mu _{ 2}^{\prime}\bar{F}(x)\quad \text{ or }\quad E(X_{t}) \leq E(Z).}$$

Definition 4.41.

Renewal new is better (worse) than renewal used, denoted by RNBRU (RNWRU), if and only if

$$\displaystyle{\mu \int _{x+t}^{\infty }\bar{F}(u)du \leq \left (\int _{ x}^{\infty }\bar{F}(u)du\right )\left (\int _{ t}^{\infty }\bar{F}(u)du\right )\!\!.}$$

Definition 4.42.

Renewal new is better (worse) than renewal used, denoted by RNBRUE (RNWRUE), if and only if E(Z _t ) ≤ E(Z), or

$$\displaystyle{2\mu \int _{x}^{\infty }\int _{ t}^{\infty }\bar{F}(u)dudt \leq \mu _{ 2}^{\prime}\int _{x}^{\infty }\bar{F}(u)du.}$$

Definition 4.43.

The random variable X has generalized increasing mean residual life (GIMRL) property if and only if, for all x ≥ 0,

$$\displaystyle{\frac{\int _{t}^{\infty }\bar{F}(u)du} {\bar{F}(t + x)} \mbox{ is increasing in $t$.}}$$

Definition 4.44.

Harmonically new renewal better than used in expectation, denoted by HRNBUE, if

$$\displaystyle{\bar{F}_{Z}(t) \leq {e}^{-\frac{t} {\mu _{Z}} }.}$$

The implications among these classes are as follows:

$$\displaystyle{ \text{GIMRL } \Rightarrow \text{ RNBU } \Rightarrow \text{ RNBUE } \Rightarrow \text{ HRNBUE}, }$$

(4.29)

$$\displaystyle{\text{NBU } \Rightarrow \text{ NBRU}.}$$

By comparing the above definitions, we see that NBRU property is the same as NBUC discussed earlier. Bon and Illayk [116] established that if the first two moments of X are finite and if X has HRNBUE property, then X has an exponential distribution. Thus, all classes implied in (4.29) are gathered in the exponential class. The conversion of Definitions 4.38–4.44 can be accomplished in the same manner as in the earlier cases.

Since we have discussed a large number of classes based on ageing concepts and given separate implications, a consolidated diagram showing all the classes and mutual implications is presented in Fig. 4.1 for a quick reference.

Fig. 4.1

Implications among different ageing concepts

4.6 Classes Based on Concepts in Reversed Time

Parallel developments have been attempted to generate life distributions based on the monotonicity properties of the reversed hazard rate function, reversed mean residual life function, and so on. However, a special feature of such criteria is that for lifetime random variables, they have monotonicity in only one direction (either decreasing or increasing). Hence, they fail to distinguish life distributions and are therefore of limited use in representing different types of ageing characteristics. But, other properties possessed by these classes could be of advantage in the analysis of data.

Definition 4.45.

A lifetime random variable is decreasing reversed hazard rate (DRHR) if and only if Λ(u)(λ(x)) is decreasing for all 0 < u < 1 (x > 0).

The quantile function corresponding to the reversed residual life, t − x | (X ≤ t), is

$$\displaystyle{Q_{u_{0}}(u) = Q(u) - Q((1 - u)u_{0}) =\int _{ u_{0}(1-u)}^{u_{0} }q(p)dp.}$$

Hence, Definition 4.45 is equivalent to saying that X is DRHR if and only if

$$\displaystyle{Q_{u_{0}}(u) \leq Q_{u_{1}}(u),\quad 0 < u_{1} \leq u_{2} < 1.}$$

Block et al. [111] have proved that there does not exist a non-negative random variable that has increasing reversed hazard rate function. A large class of distributions including those that are IHR like the Weibull, gamma, and Pareto are DRHR. The DRHR distributions are closed under the formation of coherent systems (Nanda et al. [462]).

Definition 4.46.

A life distribution has increasing reversed mean residual life time (increasing mean inactivity time) IMIT if and only if r(x) (R(u)) is increasing in x(u).

There is no non-negative random variable which has decreasing MIT over the entire domain (0, ∞). Further, DRHR ⇒ IMIT (Nanda et al. [462]). Similarly, the monotonicity of the reversed variance residual life D(u) can be studied.

Definition 4.47.

We say that X is increasing reversed variance residual life (IRVRL) if and only if v(x) (D(u)) is increasing in x(u).

Nanda et al. [462] have established that IMIT ⇒ IRVRL and if X is IRVRL, the coefficient of variation of reversed residual life cannot exceed unity. Li an Xu [393] introduced a new concept based on MTTF.

Definition 4.48.

A random life X is NBUR _h (new better than renewal used in the reversed hazard rate order) if and only if

$$\displaystyle{ \frac{F(x)} {\int _{0}^{x}\bar{F}(t)dt}}$$

increases in x ≥ 0.

It is easy to see from (4.19) that NBUR _h is the same as DMTTF. Some properties of the class discussed by them include the following:

• 1. DMTTF does not imply DMRL and DMRL does not necessarily imply DMTTF. Similarly, neither NBU nor NBUC imply DMTTF;

• 2. If X is absolutely continuous, then for any strictly increasing and concave (convex) function ϕ with ϕ(0) = 0, is also DMTTF (IMTTF);

• 3. DMTTF is not closed under the operation of mixtures;

• 4. IMTTF is not closed under convolution;

• 5. IMTTF is not closed under parallel systems;

• 6. If {P _K } is discrete DMTTF (that is, $\sum _{0}^{k-1}\bar{P}_{i}/P_{k}$ is decreasing), $\bar{H}(t)$ under a homogeneous Poisson shock model is also DMTTF. Properties 1–6 supplement those given earlier in the section on DMTTF.

4.7 Applications

One of the objectives of transforming the ageing concepts in the distribution function approach to quantile forms is to analyse lifetime data using quantile functions which do not have tractable distribution functions. We have introduced several quantile functions of this nature earlier in Chap. 3. Accordingly, applying the quantile form definitions of ageing criteria, we attempt an analysis of the ageing behaviour in these models. A second topic dealt with here is relative ageing. Ageing concepts are found to be of great use in evolving tests of hypothesis that the data come from a specific class of life distributions. We give the pertinent references at the end of the section.

4.7.1 Analysis of Quantile Functions

Govindarajulu’s distribution in (3.81) with

$$\displaystyle{Q(u) =\sigma [(\beta +1){u}^{\beta } -\beta {u}^{\beta +1}]}$$

has hazard quantile function as

$$\displaystyle{H(u) = [\sigma \beta (\beta +1{u}^{\beta -1}{(1 - u)}^{2}{]}^{-1}.}$$

Accordingly,

$$\displaystyle{ H^{\prime}(u) = \frac{u(1+\beta ) + (1-\beta )} {\sigma \beta (\beta +1){u}^{\beta }{(1 - u)}^{3}}. }$$

(4.30)

It is evident from (3.33) that H(u) is increasing for β ≤ 1 and for β > 1, H′(u) = 0 at $u_{0} = \frac{\beta -1} {\beta +1}$. Hence, X is IHR for 0 < β < 1, and BT for β > 1, with change point u ₀.

The mean residual quantile function from Sect. 3.4 is

$$\displaystyle{M(u) = \frac{\sigma } {(\beta +2)(1 - u)}\Big[2 - (\beta +1)(\beta +2){u}^{\beta } + 2\beta (\beta +2){u}^{\beta +1} -\beta (\beta +1){u}^{\beta +2}\Big]}$$

which is decreasing for β < 1. At β = 1,

$$\displaystyle{M(u) = \frac{2\sigma {(1 - u)}^{2}} {3} }$$

again decreases. But, at β = 2,

$$\displaystyle{M(u) = \frac{\sigma } {2}{(1 - u)}^{2}(1 + 3u)}$$

and so M′(u) = 0 at $u = \frac{1} {9}$. We see that M(u) is nonmonotone, being increasing in $(0, \frac{1} {9})$ and then decreasing in $(\frac{1} {9},1)$ with change point $u_{0} = \frac{1} {9}$. Thus, M(u) is of UBT shape. Notice that at β = 2, the change point of the failure rate is $u_{0} = \frac{1} {3}$ and at this value M(u) is decreasing. In the case of the refrigerator failure data studied in Sect. 3.4, the parameters are σ = 1 and $\hat{\beta }= 2.94$. We have M(u) initially increasing and then decreasing with change point $u\doteq0.2673$, while the hazard quantile function is BT shaped with change point $u_{0}\doteq0.493$. Thus, the change point occurs earlier for the mean residual quantile function. Figure 4.2 presents the shapes of the hazard quantile function for selected values of the parameters.

Fig. 4.2

Plots of hazard quantile function when (1) β = 0. 1, σ = 1 and (2) β = 2, σ = 1 for Govindarajalu’s distribution

Consider the power-Pareto distribution with its hazard quantile function as

$$\displaystyle{H(u) = {(1 - u)}^{\lambda _{2}}{[c{u}^{\lambda _{1}-1}\{\lambda _{1}(1 - u) +\lambda _{2}u\}]}^{-1}.}$$

The nature of the hazard rate function for some values of C ₁, λ ₁ and λ ₂ is exhibited in Fig. 4.3. Differentiating H(u), we see that the sign of H(u) depends on

$$\displaystyle{g(u) = -[{(\lambda _{1} -\lambda _{2})}^{2}{u}^{2} + (\lambda _{ 1} - 2\lambda _{1}^{2} + 2\lambda _{ 1}\lambda _{2})u +\lambda _{1}(\lambda _{1} - 1)].}$$

Denoting the admissible roots of g(u) = 0 by u ₁ and u ₂ with u ₁ > u ₂, we see that H(u) is decreasing when

$$\displaystyle{\lambda _{1}(1 - 4\lambda _{2}) + 4\lambda _{2}^{2} \leq 0\quad \text{ or }\quad \lambda _{ 1} = 0}$$

for all u. Further, H(u) decreases when

$$\displaystyle{\lambda _{1}(1 - 4\lambda _{2}) + 4\lambda _{2}^{2} > 0\mbox{ for all $u$ outside the interval $(u_{ 2},u_{1})$},}$$

and increases within (u ₂, u ₁). If there is only one root for H′(u) = 0, that is, $u_{1} = u_{2} = u_{0}$, then H(u) is decreasing. For λ ₂ = 0, H(u) is increasing. Summarizing the shape of H(u), we have

$$ \displaystyle\begin{array}{rcl} & & \mbox{ $X$ is DHR for $\lambda _{1} = 0$ or $\lambda _{1}(1 - 4\lambda _{2}) + 4\lambda _{2}^{2} \leq 0$}, {}\\ & & \mbox{ $X$ is IHR for $\lambda _{2} = 0$}, {}\\ \end{array} $$

and H(u) has opposite monotonicities to that in (u ₂, u ₁) where it is increasing. It can be verified that X is IHR when λ ₁ = 2, λ ₂ = 0, DHR when λ ₁ = 3, λ ₂ = 2, and non-monotonic when $\lambda _{1} =\lambda _{2} = \frac{1} {2}$. For an application to real data, we return to Sect. 3.6, where the power-Pareto distribution did provide a good fit for the data on the failure times of 20 electric carts. The hazard quantile function is

$$\displaystyle{H(u) = {(1 - u)}^{0.0967}{[1530.53{u}^{-0.7654}(0.2346(1 - u) + 0.0967u)]}^{-1}.}$$

Here, $\lambda _{1}(1 - 4\lambda _{2}) + 4\lambda _{2}^{2} > 0$ and so the hazard curve is initially increasing and then becomes BT shaped.

Fig. 4.3

Plots of hazard quantile function when (1) C = 0. 1, λ ₁ = 0. 5, λ ₂ = 0. 01; (2) C = 0. 5, λ ₁ = 2, λ ₂ = 0. 01; (3) C = 0. 01, λ ₁ = 2, λ ₂ = 0. 5; (4) C = 0. 01, λ ₁ = 0. 5, λ ₂ = 0. 5, for the power-Pareto distribution

The generalized Tukey lambda distribution of Freimer et al. [203] has its hazard quantile function as

$$\displaystyle{H(u) =\lambda _{2}{[{(1 - u)}^{\lambda _{4}} + {u}^{\lambda _{3}-1}(1 - u)]}^{-1}.}$$

The sign of H′(u) depends on the function

$$\displaystyle{g(u) =\lambda _{2}[\lambda _{4}{(1 - u)}^{\lambda _{4}-1} +\lambda _{3}{u}^{\lambda _{3}-1} + (1 -\lambda _{3}){u}^{\lambda _{3}-2}].}$$

Fig. 4.4

Plots of hazard quantile function when (1) λ ₁ = 0, λ ₂ = 100, $\lambda _{3} = -0.5$, $\lambda _{4} = -0.1$; (2) λ ₁ = 0, λ ₂ = 500, λ ₃ = 3, λ ₄ = 2; (3) λ ₁ = 0, λ ₂ = 2, λ ₃ = 10, λ ₄ = 5; (4) λ ₁ = 0, λ ₂ = 100, λ ₃ = 2, λ ₄ = 0. 5; (5) λ ₁ = 0, λ ₂ = 250, λ ₃ = 2, λ ₄ = 0. 001, for the generalized Tukey lambda distribution

The hazard quantile function can take on a wide variety of shapes as can be seen in Fig. 4.4. It is easy to see that X is IHR when λ ₂ > 0, λ ₄ > 0 and 0 < λ ₃ < 1, subject to the condition $\lambda _{1} - \frac{1} {\lambda _{2}\lambda _{3}} \geq 0$ which is required for X to have a life distribution. When λ ₄ = 0,

$$\displaystyle{g(u) =\lambda _{2}[{u}^{\lambda _{3}-2}(\lambda _{3}u + 1 -\lambda _{3})]}$$

so that H′(u) = 0 has a solution $u = \frac{\lambda _{3}-1} {\lambda _{3}}$. In this case, H(u) is BT-shaped. An exhaustive analysis using g(u) given above is difficult and for this reason we have presented above only some illustrative cases that exhibits the flexibility of H(u) to adopt to different kinds of ageing behaviour.

The generalized lambda distribution, like the Freimer et al. [203] model, has quite a flexible hazard quantile function. Recall from (3.5) that the distribution has

$$\displaystyle{H(u) =\lambda _{2}{[(1 - u)(\lambda _{3}{u}^{\lambda _{3}-1} +\lambda _{4}{(1 - u)}^{\lambda _{4}-1})]}^{-1}.}$$

We shall now take some special cases. When λ ₃ = 0, λ ₄ > 0,

$$\displaystyle{H(u) = \frac{\lambda _{2}} {\lambda _{4}{(1 - u)}^{\lambda _{4}}} }$$

and so X is IHR if λ ₂ > 0, and DHR when λ ₂ < 0. Setting λ ₄ = 0,

$$\displaystyle{H(u) = \frac{\lambda _{2}} {(1 - u){u}^{\lambda _{3}-1}}}$$

showing that X is IHR for 0 < λ ₃ < 1, and BT for λ ₃ > 1 with change point $u_{0} = \frac{\lambda _{3}-1} {\lambda _{3}}$. Finally, when λ ₃ = 2, λ ₄ = 1,

$$\displaystyle{H(u) = \frac{\lambda _{2}} {(1 - u)[2u + 1]}}$$

so that when λ ₂ > 0, H′(u) = 0 at $u = \frac{1} {4}$ and X is UBT with change point $u_{0} = \frac{1} {4}$. The shapes of the hazard function can also be seen from Fig. 4.5 for some selected choices of the four parameters.

Fig. 4.5

Plots of hazard quantile function when (1) λ ₁ = 1, λ ₂ = 100, λ ₃ = 0. 05, λ ₄ = 0. 5; (2) λ ₁ = 0, $\lambda _{2} = -1000$, λ ₃ = 0, $\lambda _{4} = -2$; (3) λ ₁ = 1, λ ₂ = 10, λ ₃ = 2, λ ₄ = 0; (4) λ ₁ = 0, $\lambda _{2} = -1000$, $\lambda _{3} = -2$, $\lambda _{4} = -1$, for the generalized lambda distribution

Finally, the van Staden–Loots model defined in (3.26), characterized by

$$\displaystyle{H(u) =\lambda _{2}(1 - u)[(1 -\lambda _{3}){u}^{\lambda _{4}-1} +\lambda _{3}{(1 - u)}^{\lambda _{4}-1}],}$$

also possesses different shapes of hazard quantile functions. Figure 4.6 presents the plot of H(u) for some selected parameter values showing different shapes.

Fig. 4.6

Plots of hazard quantile function when (1) λ ₁ = 0, λ ₂ = 0. 01, λ ₃ = 0. 5, $\lambda _{4} = -2$; (2) λ ₁ = 0, λ ₂ = 100, λ ₃ = 0. 5, λ ₄ = 10; (3) λ ₁ = 0, λ ₂ = 1, λ ₃ = 0. 6, λ ₄ = 0. 5; (4) λ ₁ = 0, λ ₂ = 0. 1, λ ₃ = 1, $\lambda _{4} = -5$, for the van Staden–Loots model

4.7.2 Relative Ageing

The role of relative ageing concepts is either to compare the ageing patterns of two units at a fixed time or to ascertain whether the same unit is ageing more positively or more negatively at different points of time.

Consider two units whose lifetimes follow the same distribution F(x), and let y be the chronological age of one unit and the other is new. Bryson and Siddiqui [122] then argued that $\frac{\bar{F}(y+x)} {\bar{F}(y)}$ is the probability that the older system will survive the same duration x given its survival up to time y as the new one with survival probability $\bar{F}(x)$. They define the specific ageing factor as

$$\displaystyle{A(x,y) = \frac{\bar{F}(y)\bar{F}(x)} {\bar{F}(x + y)},\quad x,y > 0,}$$

which compares the two survival probabilities. Note that A(x, y) > ( < 1) will mean that the older system has aged in the sense that it has less (more) probability of survival than a new unit. It is shown that increasing specific age factor

$$\displaystyle{A(y_{1},x) \geq A(y_{2},x)\quad \text{ for all }x \geq 0,\;y_{2} \geq y_{1}}$$

is equivalent to IHR.

Instead of comparing survival functions, Sengupta and Deshpande [526] made use of the failure rates h _X (x) and h _Y (x) of two lifetimes X and Y. Defining $\mathcal{H}_{G}(x) =\int _{ 0}^{x}h(t)dt$ as the cumulative hazard rate of X, they expressed relative ageing concepts as follows:

• 1. The random variable X ages faster than Y if $Z = \mathcal{H}_{G}(X)$ is IHR;

• 2. X is ageing faster than Y in average if Z is IHRA;

• 3. The random variable X is ageing faster than Y in quantile if Z is NBU.

They also obtained bounds and inequalities on $\bar{F}(x)$ in all three cases. Jiang et al. [292] dealt with unimodal hazard rates and defined the ageing intensity function as

$$\displaystyle{L(x) = \frac{xh(x)} {\mathcal{H}(x)} }$$

which, in terms of quantile functions, becomes

$$\displaystyle{l(u) = L(Q(u)) = \frac{Q(u)H(u)} {\int _{0}^{u}H(p)q(p)dp}.}$$

When X is IHR,

$$\displaystyle{H(u) \geq \frac{\int _{0}^{u}H(p)q(p)dp} {\int _{0}^{u}q(p)dp} }$$

and hence l(u) > 1. Thus, the value of l(u) quantifies the intensity of ageing. In a different setting, Abraham and Nair [13] denoted the remaining life of an old unit of lifelength X which has the same probability of survival as a new unit of age y by g(x, y). They then showed that

$$\displaystyle{g(x,y) = {\mathcal{H}}^{-1}(\mathcal{H}(x) + \mathcal{H}(y)) - x}$$

is a necessary and sufficient condition for

$$\displaystyle{P(X > g(x,y) + x\vert X > x) = P(X > y).}$$

That is, if y is the αth quantile of X, then g(x, y) is the αth quantile of the residual life distribution of X. A relative ageing factor is

$$\displaystyle{B(x,y) = {y}^{-1}g(x,y)}$$

which reveals the rate at which an old unit is losing or gaining life in relation to a new unit with identical life distribution. They showed that X is IHR if and only if B(x, y) is decreasing in x for every y > 0, and g(x, y) < y if and only if X is NBU. Theories on the quantification of ageing are still at the formative stage, but that they have interesting relationships with ageing concepts is evident from the above discussion.

For the application of the ageing concept in a real situation, it is desirable to have a test procedure. The tests are usually performed by assuming the null hypothesis that the population distribution is exponential against the alternative that it belongs to some specific ageing class, excluding exponential. As a basis for the test, one often uses moment inequalities, inequalities for survival functions, total time on test transformation (see next chapter), order statistics, stochastic orders (Chap. 8) and so on. A comprehensive survey of various tests available for this purpose has been provided by Lai and Xie [368].

The various ageing properties discussed in this chapter will be revisited in Chap. 8 in terms of stochastic orders. Some general theorems and properties established there will shed additional light on their relevance in reliability theory and other applied disciplines.