On variability of the mean remaining lifetime at random age

Abstract

In this short communication, we discuss the remaining lifetime and the mean remaining lifetime (MRL) of an item with a random age. We show that the MRL at random age is closely related to some well-known variability measures. First, we provide a decomposition result showing that the MRL at random age, similar to other variability measures, has a covariance representation. Under the proportional hazards (PH) model, we show that the MRL, depending on the parameter of proportionality, subsumes the Gini’s mean difference and the cumulative residual entropy as special cases. It is also shown that, under the PH model, the MRL can be expressed via the equilibrium distribution and the mean number of events in the generalized Pólya process.

1 Introduction

We consider a homogeneous population of statistically identical items with random generic lifetimes denoted by T. We assume that each item has been operating for some time that varies from item to item. Therefore, an item selected randomly from the population can be described by its random age (age composition) X. Furthermore, denote the remaining lifetime of the selected item by $T_{X}$. As a useful practical example, we can think about a population of used items at a warehouse. Another interpretation of these random variables is when we have only one item but its age (previous usage) is unknown and therefore can be modeled as a random variable (see Finkelstein 2008; Finkelstein and Vaupel 2015; Cha and Finkelstein 2018a; Hazra et al. 2018 for more details on the general setup). Note that the main emphasis in these studies was on the corresponding stochastic comparisons for the random variables of interest, whereas our paper is aimed at considering relevant variability measures. An important special case, which is considered in our paper as well, is when $X{\mathop {=}\limits ^{d}}T_{X}$, where ${\mathop {=}\limits ^{d}}$ denotes equality in distribution. This also describes the age composition of the stationary dynamic populations widely employed in population dynamics.

Another specific case is when $X{\mathop {=}\limits ^{d}}T$, which seems rather artificial at first sight. However, there are situations when this condition is fully justified in practice. For instance, assume that we have a population of failed items at a warehouse, and all of them are minimally repaired. This means that the initial age distribution for this population is equal to the generic life distribution in this population. Similar to the above, another interpretation is when a single item with an unknown age at failure is minimally repaired. In this case, the minimal repair is performed for each realization of this age.

Most of the results reported in the literature deal with the properties of a random age and the corresponding remaining (residual) lifetime and the stochastic comparisons between these random variables. It is worthwhile noting that this has been discussed mostly for dynamic populations when the age and the remaining lifetime depend on the chronological time t. Our setting is static and, thus, does not depend on t. Moreover, our main interest in this note is in the mean remaining lifetime (MRL) of items described by the random age. Specifically, our goal is to investigate the variability properties of this random variable, which, to the best of our knowledge, was not addressed in the literature so far. In fact, we show that the MRL, $E(T_{X})$ is closely related to some well-known variability measures.

The rest of the paper is organized as follows (note that the following description is useful for the presentation of the main body of the paper as it sets some of the main notions and briefly describes some of the important results): In Sect. 2, we first recall some preliminary results. This includes some well-known measures of variability such as Gini’s mean difference and cumulative residual entropy (CRE). In Sect. 3, we define the concept of the remaining lifetime at random age $T_{X}$. Then we show that $E(T_{X})$, the MRL at the random age, can be expressed in terms of the mean remaining life of T. Some comparison results are also established. In Sect. 3.1, we present a decomposition result showing that the MRL $E(T_{X})$ can be decomposed in terms of the averages of the generic lifetime T and the random age X. Some useful inequalities are derived in this section as well. Section 3 is devoted to the connection of the MRL $E(T_{X})$ and variability measures. We show that in the case when the generic lifetime of the item and the associated random age ‘form’ the corresponding proportional hazards model (PH), the MRL $E(T_{X})$, depending on the parameter of proportionality, reduces to the Gini’s mean difference and the CRE as special cases. The PH model is defined in a standard for modeling way as $\bar{F}^{\nu }(t)$, where F(t) is a distribution of a generic lifetime and $\nu >0$ is a coefficient of proportionality. It is also shown that under the PH model, $E(T_X)$ can be meaningfully expressed via the equilibrium distribution and the mean number of events in the generalized Pólya process (GPP).

2 Preliminary results

Various variability measures have been proposed in the literature in order to quantify the amount of dispersion in random variables. In what follows in this subsection, we recall some of them that are relevant to our discussion. Let X be a continuous random variable described by the cdf $F(x)=P(X\le x)$ and defined on the arbitrary support. The Gini’s mean difference corresponding to X, denoted by $\textrm{GMD}(X)$, is defined as

$$\begin{aligned} \textrm{GMD}(X)= E(|X_1-X_2|)=2\int F(x)\bar{F}(x)\hbox {d}x, \end{aligned}$$

(1)

where $X_1$ and $X_2$ are independent random variables distributed as X. One can easily verify that GMD(X)$=4\textrm{Cov}(X, F(X))$ (see Yitzhaki and Schechtman 2013). It was shown recently by Asadi et al. (2017) that when X is a nonnegative random variable, $\textrm{GMD}(X)$ can be also expressed in terms of MRL $m(t)=E(X-t|X>t)$ and the mean inactivity time ${\tilde{m}}(t)=E(t-X|X<t)$ of the minimum of random sample of size 2. The extended Gini mean difference $\textrm{EG}_\nu (X)$ is defined as a parametric extension of the GMD(X) in the following way:

$$\begin{aligned} \textrm{EG}_\nu (X)&=\int _{0}^{\infty }\left( \bar{F}(t)-\bar{F}^{\nu }(t)\right) \hbox {d}t\quad v>0. \end{aligned}$$

One can easily verify that $\textrm{EG}_\nu (X)$ has the following covariance representation (see Yitzhaki and Schechtman 2005).

$$\begin{aligned} \textrm{EG}_\nu (X)=\nu \left[ I(0< \nu < 1)-I(\nu >1)\right] \textrm{Cov}(X,\bar{F}^{\nu -1}(X)), \end{aligned}$$

where I(A) is an indicator function defined on the set A and $\nu$ is a parameter that ranges from 0 to infinity and determines the relative weight attributed to ‘various portions’ of probability distribution. For $\nu =2$, the extended Gini difference leads to GMD(X) (up to a constant). For more interpretations and applications of $\textrm{EG}_\nu (X)$ in economic studies based on different values of $\nu$, we refer to Yitzhaki and Schechtman (2013).

Another popular measure in the literature on information theory measure is the cumulative residual entropy (CRE) defined by Rao et al. (2004) as

$$\begin{aligned} \mathcal{E}(X)=-\int _{0}^{\infty }\bar{F}(x)\log \bar{F}(x)\hbox {d}x. \end{aligned}$$

(2)

As an alternative to the Shannon entropy measure, the cited authors argued that the CRE can be considered as a measure of uncertainty. They obtained several properties of the CRE and showed that this measure is useful in, e.g., computer vision and image processing. It was reported by Asadi (2017) that the following equality holds for a nonnegative random variable

$$\begin{aligned} \mathcal{E}(X)=\textrm{Cov}(X,\Lambda (X)), \end{aligned}$$

(3)

where $\Lambda (x)=-\log \bar{F}(x)$ is the cumulative failure rate. It was also noticed by Asadi and Zohrevand (2007) that the CRE is closely related to the mean remaining lifetime, m(t), of a nonnegative random variable X. In fact, it is always true that the CRE can be represented as $\mathcal{E}(X)=E(m(X))$.

Another useful fact is that the differential Shannon entropy of the equilibrium distribution has the following representations. Denote by $H(f_e)$ the differential Shannon entropy that corresponds to $f_e(x)=\frac{\bar{F}(x)}{\mu }$, then, based on the definition of the CRE

$$\begin{aligned} H(f_e)&= -\int _{0}^{\infty }f_{e}(x)\log f_{e}(x)\hbox {d}x\nonumber \\&=-\int _{0}^{\infty }\frac{\bar{F}(x)}{\mu }\log \frac{\bar{F}(x)}{\mu }\hbox {d}x\nonumber \\&= \frac{1}{\mu }\mathcal{E}(X)+\log \mu . \end{aligned}$$

(4)

3 The MRL and the MRL representation of $E(T_X)$

Consider a nonnegative random variable T with a cdf F and a reliability function $\bar{F}=1-F$. A random variable of interest in reliability and survival analysis studies is the residual random variable (remaining lifetime) defined as

$$\begin{aligned} T_t=(T-t|T>t), \quad t>0. \end{aligned}$$

Assuming that T denotes the lifetime of items in a homogeneous population, $T_t$ shows the remaining lifetime of an item given that the item has survived until t, $t>0$. The reliability of $T_t$ at time x is given by

$$\begin{aligned} P(T_t>x)=P(T-t>x|T>t)=\frac{\bar{F}(t+x)}{\bar{F}(t)}, \ \ t,x>0, \end{aligned}$$

provided that $\bar{F}(t)>0$. The expectation of $T_t$, known as the mean remaining lifetime (MRL), is of particular interest and is defined as

$$\begin{aligned} m(t)\equiv E(T_t)&=\int _{0}^{\infty }\frac{\bar{F}(t+x)}{\bar{F}(t)}\hbox {d}x. \end{aligned}$$

(5)

Assuming that each item in a generic population has been operating for some time that varies from item to item, then an item selected randomly from the population can be described by its random age X. Let $T_{X}$ be the remaining lifetime of the selected item at the random age X. If the cdf of X is denoted by G, then $T_X$ is distributed as $E_{X}(T_X)$ with survival function

$$\begin{aligned} P(T_X>t)&= \int _{0}^{\infty }\left( \frac{\bar{F}(t+x)}{\bar{F}(x)}\right) \hbox {d}G(x). \end{aligned}$$

The MRL corresponding to this survival function is

$$\begin{aligned} E(T_X)&=\int _{0}^{\infty }P\big (T_X>t\big )\hbox {d}t\\&=\int _{0}^{\infty }\left( \int _{0}^{\infty }\left( \frac{\bar{F}(t+x)}{\bar{F}(x)} \right) \hbox {d}G(x)\right) \hbox {d}t, \end{aligned}$$

where $E(T_X)$ refers to both random variables, which can be written as $E(T_X)=E_{T}(E_{X}(T_X))$.

The following simple result shows that $E(T_X)$ can be represented in terms of MRL m(t).

Proposition 1

Assume that m(t) is the MRL of T. Then $E(T_X)$ can be represented as

$$\begin{aligned} E(T_X)=E(m(X)). \end{aligned}$$

(6)

Proof

We have

$$\begin{aligned} E(T_X)&= \int _{0}^{\infty }\left( \int _{0}^{\infty }\left( \frac{\bar{F}(t+x)}{\bar{F}(x)}\right) \hbox {d}G(x)\right) \hbox {d}t\nonumber \\&=\int _{0}^{\infty }\left( \frac{\int _{x}^{\infty }{\bar{F}(t)\hbox {d}t}}{\bar{F}(x)}\right) \hbox {d}G(x)\nonumber \\&=\int _{0}^{\infty }m(x)\hbox {d}G(x)\nonumber \\ {}&=E(m(X)).\nonumber \end{aligned}$$

This shows that $E(T_X)$ can be represented as the expectation of m(X) with respect to the distribution G.

Corollary 1

In the special case when T and X are identically distributed (see the Sect. 1), i.e., $F(x)=G(x)$, the right-hand side of (6) reduces to the CRE $\mathcal{E}(T)$ in (2) (see Asadi and Zohrevand 2007), i.e.,

$$\begin{aligned} E(T_X)={E}(m(T))=-\int _{0}^{\infty }\bar{F}(t)\log \bar{F}(t)\hbox {d}t. \end{aligned}$$

(7)

As an illustration of (6), consider the following example.

Example 1

Assume that T belongs to the popular in reliability studies class of generalized Pareto distributions (GPD). A lifetime random variable T has GPD if its reliability function is given by

$$\begin{aligned} \bar{F}(t)=\left( \frac{b}{at+b}\right) ^{\frac{1}{a}+1} \quad t\ge 0, \ a>-1, b>0. \end{aligned}$$

(8)

This family of distributions includes, depending on the values of a, three distributions: the exponential distribution when $a\rightarrow 0$; the Pareto distribution when $a>0$; and the power distribution when $-1<a<0$. In particular, when $a=-1/2$, the distribution is uniform. It is easy to see that the MRL for the GPD is given as $m(t)=at+b$, i.e., a linear function of time. Now, using representation (6), we see that $E(T_X)$ is a linear function of E(X). That is,

$$\begin{aligned} E(T_X)&=aE(X)+b. \end{aligned}$$

The following proposition shows that, under some conditions, the MRL $E(T_X)$ is bounded by the CRE $\mathcal{E}(T)$.

Proposition 2

Let m(t) be increasing (decreasing) in t and $X\le _{st}T$. Then

$$\begin{aligned} E(T_X)\le (\ge ) \mathcal{E}(T). \end{aligned}$$

Proof

The result follows from the fact that $X\le _{st}T$ is equivalent to $E(\eta (X))\le E(\eta (T))$ for any increasing function $\eta$. Assuming that m(t) is increasing (decreasing), we get

$$\begin{aligned} E(T_X)&= \int _{0}^{\infty }m(x)\hbox {d}G(x)\\&\le (\ge ) \int _{0}^{\infty }m(x)\hbox {d}F(x)\\&=\mathcal{E}(T). \end{aligned}$$

The next proposition compares the MRL of two random lifetimes with different random ages.

Proposition 3

Denote by $T_1$ and $T_2$ two random lifetimes and by $m_1(t)$ and $m_2(t)$ the corresponding MRLs, respectively. Let $T_1$ and $T_2$ have the random ages $X_1$ and $X_2$, respectively, such that $X_1\le _{st}X_2$. If $m_1(t)\le m_2(t)$, for all t, and $m_2(t)$ $(or \ m_1(t))$ is increasing, then $E(T_{{1}_{X_1}})\le E(T_{{2}_{X_2}})$, where $T_{{i}_{X_i}}$ is the remaining lifetime for the lifetime i, $i=1,2$.

Proof

We assume that $m_2(t)$ is increasing. The proof for the case that $m_1(t)$ is increasing can be established similarly. We have

$$\begin{aligned} E(T_{{1}_{X_1}})&= \int _{0}^{\infty }m_{1}(x)\hbox {d}G_1(x)\\&\le \int _{0}^{\infty }m_2(x)\hbox {d}G_{1}(x)\\&\le \int _{0}^{\infty }m_2(x)\hbox {d}G_{2}(x)\\&={E}(T_{{2}_{X_2}}), \end{aligned}$$

where the first inequality follows from the assumption that $m_1(t)\le m_2(t)$ and the second inequality follows from the assumption that $X_1\le _{st}X_2$ and $m_2(t)$ is increasing (see Proposition 2).

Remark 1

In Proposition 3, if we assume that the random ages $X_1$ and $X_2$ are identically distributed then the result of the proposition remains valid without imposing the assumption that $m_2(t)$ (or $\ m_1(t))$ is increasing.

Remark 2

The upper record values, in a sequence of i.i.d. random variables $X_1, X_2,\ldots$, have applications in different areas of applied probability and reliability engineering (see Arnold et al. 1998). Let $X_i$’s have a common, absolutely continuous cdf F(t). Define a sequence of record times $U(n), n=1,2,\ldots ,$ as follows.

$$\begin{aligned} U(n+1)=\min \left\{ j:j>U(n),\;X_{j}>X_{U(n)}\right\} ,\;\; n\ge 1, \end{aligned}$$

with $U(1)=1$. Then, the sequence of upper record values $\{R_n, n\ge 1\}$ is defined by $R_n= X_{U(n)}, \ n\ge 1$, where $R_1=X_1$. The reliability function of $R_n$ is given as

$$\begin{aligned} \bar{F}_{n}(t)=\bar{F}(t)\sum _{x=0}^{n-1}\frac{[-\log \bar{F}(t)]^x}{x!}, \quad t>0,\ n=1,2\ldots . \end{aligned}$$

(9)

The upper records can be viewed as the maxima in a sample of random size n where n is determined by the values and the order of occurrence of the observations. From the reliability theory point of view, the nth record is just the failure time of the 1-out-of-U(n) system. If $S_n$ denotes the occurrence time of the n-th event in a non-homogeneous Poisson process (NHPP) with the mean value function $\Lambda (t)=-\log \bar{F}(t)$, then $R_n{\mathop {=}\limits ^{d}}S_n$, $n=1,2,\ldots$; see Gupta and Kirmani (1988). It is also well known that the process of minimal repairs (that restores an item to the state it has just before a failure) forms the corresponding NHPP. When $n=2$, from (9), we have

$$\begin{aligned} E(R_2)&=\int _{0}^{\infty }\left( -\bar{F}(t)\log \bar{F}(t)+\bar{F}(t)\right) \hbox {d}t\\&=E(T_X)+E(T), \end{aligned}$$

where the second equality is based on (7). This shows that when we have a population of failed identical items and repair them minimally ($T{\mathop {=}\limits ^{d}}X$ as discussed in the Sect. 1), $E(T_X)$ can be interpreted as the expectation of the sojourn time between the second and the first event in the NHPP.

Remark 3

A relevant fact that should be mentioned here is that the random variable $X + T_X$ is known in the literature as the relevation transform. The survival function of relevation transform is given by

$$\begin{aligned} \bar{G}\#\bar{F}(t)=\bar{G}(t)+\int _{0}^{t}\frac{\bar{F}(t)}{\bar{F}(x)}\hbox {d}G(x),\ \ t>0 \end{aligned}$$

(10)

(see Krakowski 1973). The reliability function (10) arises naturally in reliability theory. Consider a system with lifetime X. When the system fails at a time x, it is replaced by a unit with reliability $\bar{F}$ but with the same age as X. That is, with a unit with lifetime $T_x = (T - x|T > x)$. In this case, the lifetime of the system is $X+T_X$ with reliability (10). In the case that $F=G$, it gives the reliability function of the system under a minimal repair under which the failed system with lifetime T is replaced by a unit having the same reliability as T and with the same age. Under this setting, the reliability function of the system is given by

$$\begin{aligned} \bar{F}\#\bar{F}(t)=\bar{F}(t)-\bar{F}(t)\log \bar{F}(t) \end{aligned}$$

(11)

(see also, Navarro et al. 2019). Note that $\bar{F}\#\bar{F}(t)$ here is equivalent to the reliability function of the second upper record presented in (9). The joint reliability function of X and $T_X$ is investigated by Navarro (2022).

3.1 Decomposition of the MRL at random age

In this section, we provide a decomposition theorem regarding the MRL $E(T_X)$. Under the assumption that all integrals exist, we have the following theorem.

Theorem 1

The expectation $E(T_X)$ can be decomposed as

$$\begin{aligned} E(T_X)&=\textrm{Cov}(T,\phi (T))+ E(T-X), \end{aligned}$$

(12)

where

$$\begin{aligned} \phi (t)=\int _{0}^{t}\frac{\hbox {d}G(x)}{\bar{F}(x)}. \end{aligned}$$

Proof

Note that $E(T_X)$ can be further represented as

$$\begin{aligned} E(T_X)&= \int _{0}^{\infty }\left( \int _{0}^{\infty }\left( \frac{\bar{F}(t+x)}{\bar{F}(x)}\right) \hbox {d}G(x)\right) \hbox {d}t\nonumber \\&=\int _{0}^{\infty }\left( \int _{x}^{\infty }{\bar{F}(t)\hbox {d}t}\right) \frac{\hbox {d}G(x)}{\bar{F}(x)}\nonumber \\&=\int _{0}^{\infty }{\bar{F}(t)}\left( \int _{0}^{t}\frac{\hbox {d}G(x)}{\bar{F}(x)}\right) \hbox {d}t\nonumber \\&=\int _{0}^{\infty }{\bar{F}(t)}\phi (t)\hbox {d}t. \end{aligned}$$

(13)

Now, using integration by parts by assuming that $\lim _{t\rightarrow \infty } \phi (t)<\infty$, we have

$$\begin{aligned} \int _{0}^{\infty }{\bar{F}(t)}\phi (t)\hbox {d}t&= \int _{0}^{\infty }t\left( \phi (t)\hbox {d}F(t) -\hbox {d}G(t)\right) dt\\&=E\left[ T\phi (T)\right] -E(X)\\&=E\left[ T\phi (T)\right] -E(X)-E(T)E(\phi (T))+E(T)E(\phi (T))\\&=\textrm{Cov}(T,\phi (T))+ E(T-X), \end{aligned}$$

where the last equality follows from the fact that $E(\phi (T))=1$. This is true because we have

$$\begin{aligned} E(\phi (T))&=\int _{0}^{\infty } \phi (x)\hbox {d}F(x)\\&=\int _{0}^{\infty }\left( \int _{0}^{x}\frac{\hbox {d}G(t)}{\bar{F}(t)}\right) \hbox {d}F(x)\\&=\int _{0}^{\infty }\left( \int _{x}^{\infty }\hbox {d}F(t)\right) \frac{\hbox {d}G(x)}{\bar{F}(x)}\\&=\int _{0}^{\infty }\hbox {d}G(x) =1. \end{aligned}$$

Representation (12) can be interpreted as follows. The MRL $E(T_X)$ can be decomposed into two parts: One is the difference between the means of lifetime random variable T and random age X and the other one is the covariance between lifetime T and $\phi (T)$.

The next three simple corollaries immediately follow from this theorem.

Corollary 2

As $\phi$ is an increasing function, $\textrm{Cov}(T,\phi (T))\ge 0$, which in turn implies that $E(T_X)\ge E(T-X)$.

Corollary 3

An application of Cauchy–Schwartz inequality, and using the fact that $E(\phi (T))=1$, we obtain the following inequality.

$$\begin{aligned} E(T_X)&=\textrm{Cov}(T,\phi (T))+ E(T-X)\nonumber \\&\le \sigma _{T}\sigma _{\phi (T)}+ E(T-X), \end{aligned}$$

(14)

where $\sigma _{T}$ denotes the standard deviation of T and $\sigma _{\phi (T)}$ is the standard deviation of $\phi (T)$.

Note that, based on the fact that $E(\phi (T))=1$, we have

$$\begin{aligned} \sigma ^{2}_{\phi (T)}=E\left( \phi ^2(T)\right) -1. \end{aligned}$$

On the other hand, it can be easily shown that

$$\begin{aligned} E\left( \phi ^2(T)\right) =2E(\phi (X))=2E\left( \frac{\bar{G}(X)}{\bar{F}(X)}\right) . \end{aligned}$$

From this last equality, we conclude that if $T\ge _{st} X$, i.e., $\bar{F}(x)\ge \bar{G}(x)$, for all x, we get that $E\phi ^2(T)\le 2$. This, in turn, implies that $\sigma ^{2}_{\phi (T)}\le 1.$ Finally from this, we see that the inequality (14) can be written as

$$\begin{aligned} E(T_X)&\le \sigma _{T}+ E(T-X), \end{aligned}$$

(15)

when $T\ge _{st} X$ holds. From the result in Corollary 2 and this inequality, we conclude that, if $\bar{F}(x)\ge \bar{G}(x)$, for all x, then $E(T_X)$ is bounded as follows:

$$\begin{aligned} E(T-X)\le E(T_X)&\le \sigma _{T}+E(T-X). \end{aligned}$$

(16)

Corollary 4

It is easy to see that the unique solution to the following equation that equates densities of a random age and a random remaining lifetime

$$\begin{aligned} f_{e}\left( t\right) = \int _{0}^{\infty }\frac{f\left( t+x\right) }{\bar{F}\left( x\right) }f_{e}\left( x\right) \hbox {d}x \end{aligned}$$

(17)

is $f_{e}\left( t\right) =\frac{\overline{F}(t)}{\mu }$, where $\mu =E(T)$ and f is the density of T (see Finkelstein and Vaupel 2015). The pdf $f_e$ is the pdf of the equilibrium distribution (ED) that is also well known from the theory of renewal processes as the stationary distribution of the backward and forward recurrent times. Also, a delayed renewal process has stationary increments if and only if the distribution of the first cycle is equilibrium. Such processes are known in the literature as the stationary or equilibrium renewal processes (see Ross 1996). Note that $f(t+x)/\overline{F}x)$, as usual, defines the pdf of the remaining lifetime for the generic F(t) and deterministic age x. Assume that G(x) is the equilibrium distribution corresponding to F(x), i.e., $\hbox {d}G(x)=\frac{\bar{F}(x)}{\mu }$. Then, in (13), we have $\phi (x)=\frac{x}{\mu }$ and thus we easily see that

$$\begin{aligned} E(T_X)&=\frac{1}{\mu }\int _{0}^{\infty } x\bar{F}(x)\hbox {d}x\\&=\frac{E(T^2)}{2\mu }\\&=E(X). \end{aligned}$$

Obviously, this follows from the definition of the equilibrium distribution; however, here we have obtained it formally from the decomposition of $E(T_X)$.

4 Variability of MRL according to the PH model

In this section, we show that $E(T_X)$ is closely related to some well-known measures of variability. This will be done under the assumption that the MRL is modeled by the popular in reliability and statistics PH model. For relevant interpretation, as discussed in the Introduction, we start with the special case $F=G$, meaning that the generic lifetime and the age are distributed in accordance with the cdf F(t). This can be achieved by assuming that all items in a population are minimally repaired. Assume now that after minimal repairs, they start operating in a new environment that can be different from that before failures. It is well known that the impact of the environment is well described by the PH model. Therefore, the remaining lifetime after minimal repairs is defined accordingly (note that there can be other interpretations of the model to follow). Denoting $R_{\nu }(t)=P(T_X>t)$, in accordance with our assumption:

$$\begin{aligned} R_{\nu }(t)=\int _{0}^{\infty }\left( \frac{\bar{F}(t+x)}{\bar{F}(x)}\right) ^\nu \hbox {d}F(x), \end{aligned}$$

Denote by $E_{\nu }(T_X)$ the expectation of $T_X$. Then

$$\begin{aligned} E_{\nu }(T_X)=\int _{0}^{\infty }R_{\nu }(t)\hbox {d}t&= \int _{0}^{\infty }\left( \int _{0}^{\infty }\left( \frac{\bar{F}(t+x)}{\bar{F}(x)}\right) ^\nu \hbox {d}F(x)\right) \hbox {d}t\\&=\int _{0}^{\infty }\left( \int _{x}^{\infty }{\bar{F}^\nu (t)\hbox {d}t}\right) \frac{\hbox {d}F(x)}{\bar{F}^\nu (x)}\\&=\int _{0}^{\infty }{\bar{F}^\nu (t)}\left( \int _{0}^{t}\frac{\hbox {d}F(x)}{\bar{F}^\nu (x)}\right) \hbox {d}t\\&=\int _{0}^{\infty }{\bar{F}^\nu (t)}\left( \frac{1-\bar{F}^{1-\nu }(t)}{1-\nu }\right) \hbox {d}t\\&=\frac{1}{1-\nu }\int _{0}^{\infty }{\bar{F}^\nu (t)}\left( {1-\bar{F}^{1-\nu }(t)}\right) \hbox {d}t\\&=\frac{1}{1-\nu }\int _{0}^{\infty }\left( {\bar{F}^{\nu }(t)}-\bar{F}(t)\right) \hbox {d}t \end{aligned}$$

This expression helps us to show that the mean of $T_X$, $E_{\nu }(T_X)$ is related to the following important measures (depending on the values $\nu$):

• (i) If $\nu =n$ is an integer ($\ge 2$), then $E_{\nu }(T_X)$ can be written as the difference between the mean of X and the mean of the minimum of a random sample from X. That is,

  $$\begin{aligned} E_{n}(T_X)=\frac{E(X)-E\big (\min (X_1,\dots ,X_{n})\big )}{n-1}, \end{aligned}$$

  where $X_i$’s are i.i.d. random variables distributed as F(t).

• (ii) In the case that $\nu >0$, $\nu \not =1$, is real valued $E_{\nu }(T_X)$ is proportional to EG$_{\nu }(X)$. In other words, we have

  $$\begin{aligned} E_{\nu }(T_X)&=\frac{\textrm{EG}_\nu (X)}{\nu -1}\\&=\frac{\nu }{\nu -1} \left[ I(0< \nu < 1)-I(\nu >1)\right] \textrm{Cov}(X,\bar{F}^{\nu -1}(X)). \end{aligned}$$

  In particular, if $\nu =2$, we see that $E_{\nu }(T_X)=2GMD(X)$, where GMD(X) is the Gini’s mean difference corresponding to the distribution F(t).

• (iii) When $\nu \rightarrow 1$, by noting that

  $$\begin{aligned} \lim _{\nu \rightarrow 1}\left( \frac{1-\bar{F}^{1-\nu }(t)}{1-\nu }\right) =-\log \bar{F}(t), \end{aligned}$$

  we arrive at

  $$\begin{aligned} \lim _{\nu \rightarrow 1} E_{\nu }(T_X)&=-\int _{0}^{\infty }\bar{F}(t)\log \bar{F}(t)\hbox {d}t\\&=\mathcal{E}(X). \end{aligned}$$

  A result in Ardakani et al. (2020) shows that $0\le \mathcal{E}(X)\le \sigma$, where $\sigma$ is the standard deviation of X. This bound is a special case of the bounds in (16).

• (iv) It can be seen using (4) that $\lim _{\nu \rightarrow 1} E_{\nu }(T_X)$ can be represented in terms of the entropy of the equilibrium distribution as follows:

  $$\begin{aligned} \lim _{\nu \rightarrow 1} E_{\nu }(T_X)= \mu \big ( H(f_e)-\log \mu \big ) \end{aligned}$$

Remark 4

According to Dabrowska and Doksum (1988), the generalized odds ratio for the random variable T is defined as

$$\begin{aligned} \Lambda (t|\alpha )=\left\{ \begin{array}{ll} \frac{1}{\alpha } \frac{1-\bar{F}^{\alpha }(t)}{\bar{F}^{\alpha }(t)}, &{} \quad \textrm{for} \ {\alpha > 0}, \\ -\log \bar{F}(t), &{}\quad \textrm{for}\ {\alpha =0}. \end{array} \right. \end{aligned}$$

(18)

These authors also define a concept of the generalized odds rate, $\lambda (t|c)$, as the derivative of $\Lambda (t|c)$ with respect to t. Thus,

$$\begin{aligned} \lambda (t|\nu )=\frac{f(t)}{\bar{F}^{\alpha +1}(t)}. \end{aligned}$$

Clearly, for $\alpha =0$, we arrive at the failure rate that corresponds to the cdf F(t). If we assume now that the distribution of T belongs to the proportional hazards model, i.e., the survival function of T is $\bar{F}^{\alpha +1}(t)$ and the distribution of X is F(t), then the function $\phi (t)$ in (12), depending on $\alpha$, is

$$\begin{aligned} \phi (t|\alpha )=\int _{0}^{t} \lambda (x|\alpha )\hbox {d}x=\int _{0}^{t} \frac{f(x)}{\bar{F}^{\alpha +1}(x)}\hbox {d}x= \left\{ \begin{array}{ll} \frac{1}{\alpha } \frac{1-\bar{F}^{\alpha }(t)}{\bar{F}^{\alpha }(t)}, &{} {\alpha > 0}, \\ -\log \bar{F}(t), &{} \alpha =0, \end{array} \right. \end{aligned}$$

which is equal to the odds ratio (18). Now our results in (i)–(iv) can also be obtained using the foregoing discussion and (13) (for $\alpha =1-\nu$).

We conclude this section (and the paper) with an interesting observation showing that under the PH model described above, $E(T_X)$ can be meaningfully expressed via the equilibrium distribution and the mean number of events in the generalized Pólya process (GPP). See Cha (2014), Cha and Finkelstein (2018b) and Hashemi and Asadi (2021) for the definition and the detailed properties of this process that generalizes the NHPP taking into account its history. The GPP is defined via its stochastic intensity $\lambda _t$, i.e., it is a counting process $\{N(t), t\ge 0\}, N(0)=0$, with the set of parameters $(\lambda (t), \alpha , \beta )$, $\alpha \ge 0$, $\beta >0$, and $\lambda _t=(\alpha N(t-)+\beta )\lambda (t)$, where $N(t-)$ is the number of events preceding t (history). When $\alpha =0$ and $\beta =1$, the GPP reduces to the NHPP with the intensity function $\lambda (t)=\frac{f(t)}{\bar{F}(t)}$. It follows from Cha (2014) that

$$\begin{aligned} P(N(t)=n)=\frac{\Gamma (\beta /\alpha +n)}{\Gamma (\beta /\alpha )n!}(1-\exp \{-\alpha \Lambda (t)\})^n (\exp \{-\alpha \Lambda (t)\})^{\beta /\alpha }, \quad n=0,1,2,\ldots , \end{aligned}$$

where $\Lambda (t)=\int _{0}^{t}\lambda (s)\hbox {d}s$ and $\Gamma ( z)=\int _{0}^{\infty }x^{z-1}e^{-x}\hbox {d}x$ is the gamma function.

It can be shown using this equation (see Asadi 2023) that the mean number of events in [0, t) for the GGP with parameters $(\lambda (t), \alpha , 1)$ can be expressed as the generalized odds ratio, i.e.,

$$\begin{aligned} M(t)\equiv E(N(t))=\frac{1}{\alpha }\left( \frac{1-\bar{F}^{\alpha }(t)}{\bar{F}^{\alpha }(t)} \right) , \quad \alpha \in (0,\infty ). \end{aligned}$$

where F(t) is the cdf defined by the failure rate $\lambda (t)$. It directly follows from the previous remark and Eq. (13) that

$$\begin{aligned} E(T_X)&=\int _{0}^{\infty }{\bar{F}(t)}M(t)\hbox {d}t \\&=\mu \int _{0}^{\infty }M(t) f_e(t)\hbox {d}t \end{aligned}$$

where $f_e(x)=\frac{\bar{F}(x)}{\mu }$ is the pdf of the equilibrium distribution. This shows that, under the proportional hazards setting with parameter $\alpha$, $E(T_{X})$ can be represented as the expectation (with respect to the equilibrium distribution) of the mean number of events in the GGP.