On stochastic comparisons of minimum order statistics from the location–scale family of distributions

Abstract

We consider stochastic comparisons of minimum order statistics from the location–scale family of distributions that contain most of the popular lifetime distributions. Under certain assumptions, we show that the minimum order statistic of one set of random variables dominates that of another set of random variables with respect to different stochastic orders. Furthermore, we illustrate our results using some well-known specific distributions.

1 Introduction and preliminaries

The concept of order statistics has been intensively used during the last few decades as a popular tool in various areas of probability and statistics (see  Balakrishnan and Rao 1998a, b). For example, the maximum order statistic is often applied in stochastic modeling of floods and of other meteorological phenomena, whereas the minimum order statistic can be helpful in reliability and survival analysis, etc. (see, e.g., Hazra 2016; Finkelstein 2008; Finkelstein and Cha 2013).

Let $\{X_1,X_2,\ldots ,X_n\}$ be a collection of random variables, and $X_{1:n}\le X_{2:n}\le \cdots \le X_{n:n}$ be the corresponding order statistics. For $1\le k\le n$, $X_{k:n}$ is called the kth order statistic. Suppose that the set of random variables $\{X_1,X_2,\ldots ,X_n\}$ represents the lifetimes of components of a system. Then the reliability (survival) function that characterizes a k-out-of-n system formed by $\{X_1,X_2,\ldots ,X_n\}$ is the same as that for $X_{n-k+1:n}$. Thus, to study a k-out-of-n system, it is sufficient to deal with the $(n-k+1)$th order statistic and vice versa, which is very important in reliability analysis.

Stochastic comparisons of different order statistics have been intensively studied in the literature (see, e.g., Balakrishnan and Zhao 2013a, and references therein). However, most of the corresponding results are obtained with respect to specific distributions, for example, exponential, Weibull, gamma, generalized exponential, etc. (see  Bon and Păltănea 2006; Zhao et al. 2009; Zhao and Balakrishnan 2011, 2012; Balakrishnan and Zhao 2013b; Torrado 2015a; Torrado and Veerman 2012; Gupta et al. 2015; Torrado 2015b, and Kundu et al. 2016 to name a few). Much less studies have been reported with respect to general families of distributions, namely, the location–scale family, the proportional hazard rates family, etc. (see Pledger and Proschan 1971; Proschan and Sethuraman 1976; Kochar and Xu 2007; Khaledi et al. 2011; Li and Li 2016 and references therein). It should be noted that a special attention has to be paid to the location–scale family as it contains most of the popular lifetime distributions. Some relevant results for the specific case of the scale family of distributions could be found in Khaledi et al. (2011), Li et al. (2016), Kochar and Torrado (2015), Li and Li (2016), Ding et al. (2017), Torrado (2017), and Fang and Li (2017). However, stochastic comparisons of order statistics when location and scale parameters are different for different sets of random variables have not been considered in the literature so far. This is a major challenge to be addressed in the present paper.

Specifically, we investigate stochastic comparison of two minimum order statistics formed from two different sets of random variables having different/the same location and scale parameters. Our general results are obtained for the location–scale family of distributions. Furthermore, as specific illustrative cases, we consider a number of well known lifetime distributions as baseline for the scale–location family. For each type of the discussed stochastic comparisons, we present important practical results defining baseline distributions that comply with our comparisons. We also prove a number of meaningful supplementary propositions in order to obtain these results.

Thus, our paper develops the theory of stochastic comparisons for two minimum order statistics that are formed from two different sets of random variables having different/the same location as well as different/the same scale parameters. In what follows in this section, we provide some notation and basic definitions.

For an absolutely continuous random variable Y, we denote the probability density function (pdf) by $f_Y(\cdot )$, the cumulative distribution function (cdf) by $F_Y(\cdot )$, and the hazard (failure) rate function by $r_Y(\cdot )$. The survival or reliability function of a random variable Y is written as ${\bar{F}}_Y(\cdot )=1-F_Y(\cdot )$. We denote the set of real numbers by $\mathbb {R}$.

A random variable X is said to follow the location–scale family, written as $X\sim$ LS$(\lambda ,\theta )$, if its distribution function is represented as

$$\begin{aligned} F_X(x;\lambda ,\theta )=F\left( \frac{x-\lambda }{\theta }\right) ,\quad x> \lambda , \end{aligned}$$

where $\lambda$ ($\in \mathbb {R}$) and $\theta \;(>0)$ are the location and the scale parameters, respectively, and $F(\cdot )$ is the baseline distribution function (cf. Marshall and Olkin 2007). We also denote the pdf and the hazard rate function of the baseline distribution by $f(\cdot )$ and $r(\cdot )$, respectively. A list of well known distributions that could be considered as baseline is given in Table 1.

Table 1 A list of baseline distributions

Majorization orders are quite useful for establishing various inequalities. In the literature, different kinds of majorization orders have been developed in order to study various types of problems in mathematics, statistics, economics and other disciplines. For definitions of different majorization orders, a reader may refer to Marshall et al. (2011) and Balakrishnan and Zhao (2013a). For any two vectors ${\varvec{x}}\in I^n$ and ${\varvec{y}}\in I^n$ ( where $I\subseteq \mathbb {R}$), we write ${\varvec{x}}{\mathop {\succeq }\limits ^{m}}{\varvec{y}}$ (resp. ${\varvec{x}}{\mathop {\succeq }\limits ^{w}}{\varvec{y}}$, ${\varvec{x}}{\succeq }_{w}\;{\varvec{y}}$) to denote the majorization order (resp. weakly supermajorization order, weakly submajorization order) between ${\varvec{x}}$ and ${\varvec{y}}$.

Stochastic ordering is an important tool for comparison of random variables. Various stochastic orders have been studied in the literature. For definitions of different stochastic orders, a reader may refer to Shaked and Shanthikumar (2007), and Li and Li (2016). For any two nonnegative random variables X and Y, we write $X\le _{st}Y$ (resp. $X\le _{hr}Y$, $X\le _{disp}Y$, $X\le _{lr}Y$, $X\le _{R-HR}Y$) to denote that X is smaller than Y with respect to the usual stochastic order (resp. hazard rate order, dispersive order, likelihood ratio order, ageing faster order in terms of hazard rate).

As usual, throughout the paper, increasing and decreasing mean non-decreasing and non-increasing, respectively. By $a{\mathop {=}\limits ^{\text {sgn}}}b$ and $a{\mathop {=}\limits ^{\text {def.}}}b$, we mean that a and b have the same sign, and b is defined as a, respectively. For convenience of notation, we also write $\lambda _{\min }=\min \{\lambda _1,\lambda _2,\ldots ,\lambda _n\}$. We use bold symbols to represent vectors, namely, ${\varvec{\lambda }}=(\lambda _1,\lambda _2,\ldots ,\lambda _n)$, ${\varvec{\lambda }}^{-1}=(\lambda _1^{-1},\lambda _2^{-1},\ldots ,\lambda _n^{-1})$ and ${{\varvec{e^\lambda }}}=(e^{\lambda _1},e^{\lambda _2},\ldots ,e^{\lambda _n})$. We also use the following two notations: $\mathcal {D}=\{\left( x_{1},x_2,\ldots ,x_{n}\right) \in \mathbb {R}^n: \;x_{1}\ge x_2\ge \cdots \ge x_{n}\}$ and $\mathcal {E}=\{\left( x_{1},x_2,\ldots ,x_{n}\right) \in \mathbb {R}^n: \;x_{1}\le x_2\le \cdots \le x_{n}\}$.

The rest of the paper is organized as follows. In Sect. 2, we compare two minimum order statistics. We show that, if the set of location/scale parameters of one set of random variables majorizes that of another set of random variables, then the minimum order statistic of the first set of random variables is dominated by that of the other set of random variables with respect to different stochastic orders. In Sect. 3, we discuss some practical results showing for which baseline distributions (and the corresponding admissible regions of parameters) our proposed results hold. The concluding remarks are given in Sect. 4.

For convenience, all proofs of theorems, lemmas, and propositions have been deferred to the “Appendix”.

2 Main results

In this section, we discuss some stochastic comparisons for two minimum order statistics $X_{1:n}$ and $Y_{1:n}$, which are formed by two sets of independent random variables $\{X_1,X_2,\ldots ,X_n\}$ and $\{Y_1,Y_2,\ldots ,Y_n\}$, respectively. We consider the following models for the location scale (LS) families:

• (M1) $X_i\sim$ LS$(\lambda _i,\theta _i)$ and $Y_i\sim$ LS$(\mu _i,\theta _i)$,    for $i=1,2,\ldots ,n$.

• (M2) $X_i\sim$ LS$(\lambda _i,\theta )$ and $Y_i\sim$ LS$(\mu _i,\theta )$,    for $i=1,2,\ldots ,n$.

• (M3) $X_i\sim$ LS$(\lambda _i,\theta _i)$ and $Y_i\sim$ LS$(\lambda _i,\delta _i)$,    for $i=1,2,\ldots ,n$.

• (M4) $X_i\sim$ LS$(\lambda ,\theta _i)$ and $Y_i\sim$ LS$(\lambda ,\delta _i)$,    for $i=1,2,\ldots ,n$.

• (M5) $X_i\sim$ LS$(\lambda _i,\theta _i)$ and $Y_i\sim$ LS$(\mu _i,\delta _i)$,    for $i=1,2,\ldots ,n$.

• (M6) $X_i\sim$ LS$(\lambda ,\theta _i)$ and $Y_i\sim$ LS$(\lambda ,\theta )$,    for $i=1,2,\ldots ,n$.

• (M7) $X_i\sim$ LS$(\lambda ,\theta _1)$ and $Y_i\sim$ LS$(\lambda ,\delta _1)$,    for $i=1,2,\ldots ,n_1$, $X_j\sim$ LS$(\lambda ,\theta _2)$ and $Y_j\sim$ LS$(\lambda ,\delta _2)$,    for $j=n_1+1,n_1+2,\ldots ,n_1+n_2(=n)$.

• (M8) $X_i\sim$ LS$(\lambda ,\theta _1)$ and $Y_i\sim$ LS$(\lambda ,\delta _1)$,    for $i=1,2,\ldots ,n_1$, $X_j\sim$ LS$(\lambda ,\alpha )$ and $Y_j\sim$ LS$(\lambda ,\alpha )$,    for $j=n_1+1,n_1+2,\ldots ,n_1+n_2(=n)$.

In what follows, we use the following assumptions:

• (A1) $\{{\varvec{\lambda }}\in \mathcal {E},\; {\varvec{\mu }}\in \mathcal {E},\;{\varvec{\theta }}\in \mathcal {E}\}$ or $\{{\varvec{\lambda }}\in \mathcal {D},\;{\varvec{\mu }}\in \mathcal {D},\;{\varvec{\theta }}\in \mathcal {D}\}$.

• (A2) $\{{\varvec{\lambda }}\in \mathcal {E},\; {\varvec{\theta }}\in \mathcal {E},\;{\varvec{\delta }}\in \mathcal {E}\}$ or $\{{\varvec{\lambda }}\in \mathcal {D},\;{\varvec{\theta }}\in \mathcal {D},\;{\varvec{\delta }}\in \mathcal {D}\}$.

• (A3) $\{{\varvec{\lambda }}\in \mathcal {E},\;{\varvec{\mu }}\in \mathcal {E},\; {\varvec{\theta }}\in \mathcal {E},\;{\varvec{\delta }}\in \mathcal {E}\}$ or $\{{\varvec{\lambda }}\in \mathcal {D},\;{\varvec{\mu }}\in \mathcal {D},\;{\varvec{\theta }}\in \mathcal {D},\;{\varvec{\delta }}\in \mathcal {D}\}$.

• (A4) $\left\{ (\theta _1,\theta _2)\in \mathcal {E}, (\delta _1,\delta _2)\in \mathcal {E},(n_1,n_2)\in \mathcal {D}\right\}$ or $\{(\theta _1,\theta _2)\in \mathcal {D}, (\delta _1,\delta _2)\in \mathcal {D}, (n_1,n_2)\in \mathcal {E}\}$

• (A5) $\{\theta _1\ge \delta _1\ge \delta _2 \ge \theta _2, n_1\le n_2 \}$ or $\{\theta _1\le \delta _1\le \delta _2 \le \theta _2, n_1\ge n_2 \}$.

• (A6) $1/\theta _1\ge \max \{1/\delta _1,1/\alpha \}$.

Furthermore, in what follows, we also use the following conditions:

• (C1) r(u) is increasing in $u>0$.

• (C2) r(u) is decreasing in $u>0$.

• (C3) r(u) is convex in $u>0$.

• (C4) ur(u) is increasing in $u>0$.

• (C5) ur(u) is convex in $u>0$.

• (C6) ur(u) is concave in $u>0$.

• (C7) $ur'(u)/{r(u)}$ is decreasing in $u>0$.

• (C8) $ur'(u)/{r(u)}$ is concave in $u>0$.

2.1 Usual stochastic order

In this subsection, we consider $X_{1:n}$ and $Y_{1:n}$ from two different sets of random variables having different/the same location as well as different/the same scale parameters. We compare these statistics with respect to the usual stochastic order.

In the following theorem, we consider two minimum order statistics that are formed from two different sets of random variables having different sets of location parameters but the same set of scale parameters. We show that the minimum order statistic of one set of random variables dominates that of the other set provided that the set of location parameters of the first set is weakly supermajorized by that of the other set. We remind that all proofs are deferred to the “Appendix”, where applicable.

Theorem 2.1

If (M1), (A1) and (C1) hold, then ${{\varvec{\lambda }}} {\mathop {\succeq }\limits ^{w}}{{\varvec{\mu }}} \text { implies } X_{1:n}\le _{st} Y_{1:n}.$

If the scale parameters in the above theorem are equal, then this result can be proved without assumption (A1) as given in the following theorem. The proof, being similar, is omitted.

Theorem 2.2

If (M2) and (C1) hold, then ${{\varvec{\lambda }}} {\mathop {\succeq }\limits ^{w}}{{\varvec{\mu }}} \text { implies } X_{1:n}\le _{st} Y_{1:n}.$

In the following theorem, we show that the same result, as in Theorems 2.1 and 2.2, holds under the assumption that both sets of random variables have the same set of location parameters but different sets of scale parameters. The proof could be performed in the same line as in Theorem 2.1.

Theorem 2.3

If (M3), (A2) and (C1) hold, then ${{{\varvec{\theta }}}}^{-1} \succeq _{w} {{{\varvec{\delta }}}}^{-1} \text { implies } X_{1:n}\le _{st} Y_{1:n}.$

The next theorem shows that, if all location parameters are equal, then the assumption (A2) given in Theorem 2.3 could be relaxed. The proof is given in Torrado (2017).

Theorem 2.4

If (M4) and (C1) (resp. (C2)) hold, then

$$\begin{aligned} {{{\varvec{\theta }}}}^{-1} \succeq _{w}(\text {resp. }{\mathop {\succeq }\limits ^{w}}) \;{{{\varvec{\delta }}}}^{-1} \text { implies } X_{1:n}\le _{st} \;(\text {resp. }\ge _{st})\;Y_{1:n}. \end{aligned}$$

The next result immediately follows from Theorems 2.1 and 2.3. We assume that location and scale parameters are different for different sets of random variables.

Theorem 2.5

If (M5), (A3) and (C1) hold, then ${{\varvec{\lambda }}} {\mathop {\succeq }\limits ^{w}}{{\varvec{\mu }}} \text { and }{{{\varvec{\theta }}}}^{-1} \succeq _{w} {{{\varvec{\delta }}}}^{-1} \text { imply } X_{1:n}\le _{st} Y_{1:n}.$

2.2 Hazard rate order and dispersive order

In this subsection, we compare two minimum order statistics with respect to the hazard rate order and the dispersive order.

In the following theorem, we show that the minimum order statistic of one set of random variables dominates that of another set with respect to the hazard rate order provided that the set of location parameters of the first set is weakly supermajorized by that of the other. We also assume that both sets of random variables have the same set of scale parameters.

Theorem 2.6

If (M1), (A1), (C1) and (C3) hold, then ${{\varvec{\lambda }}} {\mathop {\succeq }\limits ^{w}}{{\varvec{\mu }}} \text { implies } X_{1:n}\le _{hr} Y_{1:n}.$

If the scale parameters in the above theorem are equal, then the result can be proved without assumption (A1) as given in the following theorem. The proof, being similar, is omitted.

Theorem 2.7

If (M2), (C1) and (C3) hold, then ${{\varvec{\lambda }}} {\mathop {\succeq }\limits ^{w}}{{\varvec{\mu }}} \text { implies } X_{1:n}\le _{hr}Y_{1:n}.$

In the next theorem we assume that two sets of random variables have the same set of location parameters but different sets of scale parameters. The proof could be performed in the same line as in Theorem 2.6.

Theorem 2.8

If (M3), (A2), (C4) and (C5) hold, then ${{{\varvec{\theta }}}}^{-1} \succeq _{w} {{{\varvec{\delta }}}}^{-1} \text { implies } X_{1:n}\le _{hr} Y_{1:n}.$

The following theorem shows that, if all location parameters are equal, then the result given in Theorem 2.8 could be proven without assumption (A2). The proofs of (i) and (iii) are given in Khaledi et al. (2011), whereas the proof of (ii) is given in Torrado (2017). The proof of (iv) is omitted.

Theorem 2.9

Suppose that (M4) holds. The following results are true:

(i):

If (C5) (resp. (C6)) holds, then ${{{\varvec{\theta }}}}^{-1} {\mathop {\succeq }\limits ^{m}}{{{\varvec{\delta }}}}^{-1} \text { implies }X_{1:n}\le _{hr} (\text {resp. } \ge _{hr} )\;Y_{1:n}.$

(ii):

If (C4) and (C5) (resp. (C6)) hold, then ${{{\varvec{\theta }}}}^{-1} \succeq _{w} (\text {resp. }{\mathop {\succeq }\limits ^{w}})\;{{{\varvec{\delta }}}}^{-1} \text { implies } X_{1:n}\le _{hr} (\text {resp. } \ge _{hr} )\;Y_{1:n}.$

(iii):

If (C2) and (C6) hold, then ${{{\varvec{\theta }}}}^{-1} {\mathop {\succeq }\limits ^{m}}{{{\varvec{\delta }}}}^{-1} \text { implies } X_{1:n} \ge _{disp} Y_{1:n}.$

(iv):

If (C2), (C4) and (C6) hold, then ${{{\varvec{\theta }}}}^{-1} {\mathop {\succeq }\limits ^{w}}{{{\varvec{\delta }}}}^{-1} \text { implies } X_{1:n} \ge _{disp} Y_{1:n}.$

In the next theorem, we assume that the location and scale parameters are different for different sets of random variables. The proof follows from Theorems 2.6 and 2.8 by using the fact that ‘r(u) is increasing and convex in u’ implies ’ur(u) is increasing and convex in u’.

Theorem 2.10

If (M5), (A3), (C1) and (C3) hold, then ${{\varvec{\lambda }}} {\mathop {\succeq }\limits ^{w}}{{\varvec{\mu }}} \text { and }{{{\varvec{\theta }}}}^{-1} \succeq _{w} {{{\varvec{\delta }}}}^{-1} \text { imply } X_{1:n}\le _{hr}Y_{1:n}.$

2.3 Ageing faster order in terms of hazard rate

In this subsection, we discuss some stochastic comparisons for minimum order statistics with respect to the R-HR order.

Due to mathematical complexity, we cannot approach the general case as in the previous two subsections. Therefore, we consider a particular case called the ‘multiple-outlier model’. In this model, it is assumed that a set of random variables can be written as a union of two subsets, where all random variables within each subset are identically distributed. Stochastic comparisons for different multiple-outlier models (namely, multiple-outlier proportional hazard rates model, multiple-outlier Weibull model, etc.) have been studied in the literature (see Balakrishnan and Zhao 2013a, and the reference therein) for some specific cases. In this paper, we focus on a rather general multiple-outlier location–scale model. In the next theorem we show that under certain assumptions the minimum order statistic of one set of random variables dominates that of the other set with respect to the R-HR order.

Theorem 2.11

If (M7), (A4), (C4), (C7) and (C8) hold, then

$$\begin{aligned}&\left( \underbrace{{\theta _1}^{-1},\ldots ,{\theta _1}^{-1},}_{n_1} \underbrace{{\theta _2}^{-1},\ldots ,{\theta _2}^{-1}}_{n_2}\right) \\&\quad {\mathop {\succeq }\limits ^{m}}\left( \underbrace{{\delta _1}^{-1},\ldots ,{\delta _1}^{-1},}_{n_1} \underbrace{{\delta _2}^{-1},\ldots ,{\delta _2}^{-1}}_{n_2}\right) \text { implies }X_{1:n}\ge _{R-HR}Y_{1:n}. \end{aligned}$$

The next theorem shows that the assumption (A4) given in Theorem 2.11 could be relaxed when $n_1=n_2=1$. The proof is similar to that of Theorem 2.11 and hence omitted.

Theorem 2.12

Suppose that $n_1=n_2=1$. If (M7), (C4), (C7) and (C8) hold, then $\left( {\theta _1}^{-1},{\theta _2}^{-1}\right)$ ${\mathop {\succeq }\limits ^{m}}\left( {\delta _1}^{-1},{\delta _2}^{-1}\right) \text { implies } X_{1:2}\ge _{R-HR}Y_{1:2}.$

The following theorem shows that, if we impose some additional restrictions on parameters, then the result given in Theorem 2.11 holds under a weaker condition. The proof is similar to that of Theorem 4.5 of Kochar and Torrado (2015) and hence omitted.

Theorem 2.13

If (M8), (A6), (C4) and (C7) hold, then

$$\begin{aligned}&\left( \underbrace{{\theta _1}^{-1},\ldots ,{\theta _1}^{-1},}_{n_1} \underbrace{{\alpha }^{-1},\ldots ,{\alpha }^{-1}}_{n_2}\right) \\&\quad \succeq _{w}\left( \underbrace{{\delta _1}^{-1},\ldots ,{\delta _1}^{-1},}_{n_1} \underbrace{{\alpha }^{-1},\ldots ,{\alpha }^{-1}}_{n_2}\right) \text { implies }X_{1:n}\ge _{R-HR}Y_{1:n}. \end{aligned}$$

The next theorem is a generalization of Theorems 2.11 and 2.13.

Theorem 2.14

If (M7), (A5), (C4), (C7) and (C8) hold, then

$$\begin{aligned}&\left( \underbrace{{\theta _1}^{-1},\ldots ,{\theta _1}^{-1},}_{n_1} \underbrace{{\theta _2}^{-1},\ldots ,{\theta _2}^{-1}}_{n_2}\right) \\&\quad \succeq _{w}\left( \underbrace{{\delta _1}^{-1},\ldots ,{\delta _1}^{-1},}_{n_1} \underbrace{{\delta _2}^{-1},\ldots ,{\delta _2}^{-1}}_{n_2}\right) \text { implies }X_{1:n}\ge _{R-HR}Y_{1:n}. \end{aligned}$$

Two minimum order statistics, one from a heterogeneous population and the other from a homogeneous population, are compared in the following theorem, which may be found in Li and Li (2016). However, the set of sufficient conditions given here is different and what is important, the current conditions are often easier to check.

Theorem 2.15

If (M6), (C4), (C7) and (C8) hold, then $n{\theta }^{-1}\le \sum \nolimits _{i=1}^{n}{\theta _i}^{-1}\text { implies } X_{1:n}\ge _{R-HR}Y_{1:n}.$

2.4 Likelihood ratio order

In this subsection, we study some stochastic comparisons for minimum order statistics with respect to the likelihood ratio order. In the following theorem, we compare two minimum order statistics, which are formed from two different multiple-outlier location–scale models, with respect to the likelihood ratio order. The proof follows from Theorem 1.C.4 of Shaked and Shanthikumar (2007), and Theorems 2.9(i) and 2.11.

Theorem 2.16

If (M7), (A4), (C4), (C5), (C7) and (C8) hold, then

$$\begin{aligned}&\left( \underbrace{{\theta _1}^{-1},\ldots ,{\theta _1}^{-1},}_{n_1} \underbrace{{\theta _2}^{-1},\ldots ,{\theta _2}^{-1}}_{n_2}\right) \\&\qquad {\mathop {\succeq }\limits ^{m}}\left( \underbrace{{\delta _1}^{-1},\ldots ,{\delta _1}^{-1},}_{n_1} \underbrace{{\delta _2}^{-1},\ldots ,{\delta _2}^{-1}}_{n_2}\right) \text { implies }X_{1:n}\le _{lr}Y_{1:n}. \end{aligned}$$

The next theorem shows that assumption (A4) given in Theorem 2.16 could be relaxed when $n_1=n_2=1$. The proof follows from Theorem 1.C.4 of Shaked and Shanthikumar (2007), and Theorems 2.9(i) and 2.12.

Theorem 2.17

Suppose that $n_1=n_2=1$. If (M7), (C4), (C5), (C7) and (C8) hold, then $\left( {\theta _1}^{-1},{\theta _2}^{-1}\right) {\mathop {\succeq }\limits ^{m}}\left( {\delta _1}^{-1},{\delta _2}^{-1}\right) \text { implies } X_{1:2}\le _{lr}Y_{1:2}.$

In the following theorem, we show the same result as in Theorem 2.16, but under some weaker condition. The proof follows from Theorem 1.C.4 of Shaked and Shanthikumar (2007), and Theorems 2.9(ii) and 2.13.

Theorem 2.18

If (M8), (A6), (C4), (C5) and (C7) hold, then

$$\begin{aligned}&\left( \underbrace{{\theta _1}^{-1},\ldots ,{\theta _1}^{-1},}_{n_1} \underbrace{{\alpha }^{-1},\ldots ,{\alpha }^{-1}}_{n_2}\right) \\&\qquad \succeq _{w}\left( \underbrace{{\delta _1}^{-1},\ldots ,{\delta _1}^{-1},}_{n_1} \underbrace{{\alpha }^{-1},\ldots ,{\alpha }^{-1}}_{n_2}\right) \text { implies }X_{1:n}\le _{lr}Y_{1:n}. \end{aligned}$$

The next theorem is a generalization of Theorems 2.16 and 2.18. The proof could be done in the same line as in Theorem 2.14, and hence omitted.

Theorem 2.19

If (M7), (A5), (C4), (C5), (C7) and (C8) hold, then

$$\begin{aligned}&\left( \underbrace{{\theta _1}^{-1},\ldots ,{\theta _1}^{-1},}_{n_1} \underbrace{{\theta _2}^{-1},\ldots ,{\theta _2}^{-1}}_{n_2}\right) \\&\qquad \succeq _{w}\left( \underbrace{{\delta _1}^{-1},\ldots ,{\delta _1}^{-1},}_{n_1} \underbrace{{\delta _2}^{-1},\ldots ,{\delta _2}^{-1}}_{n_2}\right) \text { implies }X_{1:n}\le _{lr}Y_{1:n}. \end{aligned}$$

Two minimum order statistics, one from a heterogeneous population and the other from a homogeneous population, are compared in the following theorem. The proof follows from Theorem 1.C.4 of Shaked and Shanthikumar (2007), and Theorems 2.9(i) and 2.15.

Theorem 2.20

If (M6), (C4), (C5), (C7) and (C8) hold, then $n{\theta }^{-1}\le \sum \nolimits _{i=1}^{n}{\theta _i}^{-1}\text { implies } X_{1:n}\le _{lr}Y_{1:n}.$

3 Applications

In Table 1, we have listed a number of lifetime distributions that could be considered as the baseline distribution for a location–scale family. We discuss now different distributional properties of all these distributions. These properties will be used for the meaningful illustration of the main results of this paper. We begin with the following lemma, which will be used in the next proposition. The proofs of (iii) and (iv) are omitted.

Lemma 3.1

Let $Z\sim$FP($\alpha ,\beta ,\gamma$). Then, the following results are true:

$(\mathrm{i})$ :

$u\frac{r_Z'(u)}{r_Z(u)}\ge -\,1$, for all $u>0$, $\alpha>0,\beta >0$ and $\gamma >0$;

$(\mathrm{ii})$ :

$u\frac{r_Z'(u)}{r_Z(u)}\le -\,1+\frac{\beta }{\gamma }$, for all $u>0$, $\alpha>0,\beta >1$ and $\gamma >0$;

$(\mathrm{iii})$ :

$\lim \limits _{u \rightarrow 0}ur_Z(u)=0$ and $\lim \limits _{u \rightarrow \infty }u r_Z(u)=\frac{\alpha }{\gamma }$;

$(\mathrm{iv})$ :

$\lim \limits _{u \rightarrow 0}u\frac{r_Z'(u)}{r_Z(u)}=-\,1+\frac{\beta }{\gamma }$ and $\lim \limits _{u \rightarrow \infty } u\frac{ r_Z'(u)}{r_Z(u)}=-1$.

Note that, in the discussion to follow, the location and the scale parameters without any loss of generality are taken as zero and one, respectively. This is because all baseline distributions discussed here will be properly analysed when location and/or scale parameters are introduced, although the transformed distributions may not belong to the parent family. In the following proposition, we discuss some distributional properties of the Feller–Pareto distribution. The proofs of (i) and (ii) follow from Lemma 3.1.

Proposition 3.1

Let $Z\sim$FP($\alpha ,\beta ,\gamma$). Then, the following results are true:

$(\mathrm{i})$ :

(C4) holds, for all $\alpha ,\beta ,\gamma >0$;

$(\mathrm{ii})$ :

(C2) holds, for all $1<\beta \le \gamma$ and $\alpha >0$;

$(\mathrm{iii})$ :

(C7) holds, for all $0<\alpha < 1\le \beta$ and $\gamma >0$.

In the next proposition, we discuss some properties of the generalized Pareto distribution. The proofs are omitted.

Proposition 3.2

Let $Z\sim$GP($\xi$). Then (C2), (C4), (C6) and (C7) hold, for all $\xi >0$.

Some properties of the Burr distribution are discussed in the following proposition. The proofs are omitted.

Proposition 3.3

Let $Z\sim$Burr(c, k). Then (C2), (C4), (C6) and (C7) hold, for all $0< c \le 1$ and $k>0$.

In the next proposition, we discuss some properties of the exponentiated Weibull distribution. The proof of (i) could be found in Mudholkar et al. (1995), whereas the proof of (iii) is given in Torrado (2017); the other proofs are omitted.

Proposition 3.4

Let $Z\sim$EW($\alpha ,\beta$). Then, the following results are true:

$(\mathrm{i})$ :

(C2) holds, for all $\alpha \le 1$ and $\alpha \beta \le 1$, and (C1) holds, for all $\alpha \ge 1$ and $\alpha \beta \ge 1$;

$(\mathrm{ii})$ :

(C3) holds, for $\{0<\alpha \le 1,\beta =1\}$ or $\{\alpha \ge 2, \beta =1\}$;

$(\mathrm{iii})$ :

(C4) holds, for all $\alpha >0$ and $\beta \ge 1$;

$(\mathrm{iv})$ :

(C5) holds, for all $\alpha \ge 1$ and $\beta =1$, and (C6) holds, for all $\alpha \le 1$ and $\beta =1$;

$(\mathrm{v})$ :

$u{ r_Z'(u)}/{ r_Z(u)}$ is constant with respect to u, for $\alpha >0$ and $\beta =1$.

Some properties of the power-generalized Weibull distribution are discussed in the next proposition. The proofs of (i) and (ii) are given in Khaledi et al. (2011), whereas the proof of (iv) could be found in Ding et al. (2017). The other proofs are omitted.

Proposition 3.5

Let $Z\sim$PGW(c, k). Then, the following results are true:

$(\mathrm{i})$ :

(C4) holds, for all $c>0$ and $k>0$;

$(\mathrm{ii})$ :

(C2) and (C6) hold, for $c\le k$ and $c<1$, and (C1) and (C5) hold, for $c> k$ and $c> 1$;

$(\mathrm{iii})$ :

(C3) holds, for $1/2\le c\le k\le 1$ or $c\le 1/2, k\ge 1$ or $c\ge 2, k\le 1$;

$(\mathrm{iv})$ :

(C7) holds, for $c>0$ and $k\ge 1$;

$(\mathrm{v})$ :

(C8) holds, for $\{0<c\le 1,0<k\le 1\}$ or $\{c\ge 3, k\ge 1\}$.

The following proposition discusses some properties of the exponentiated Weibull-geometric distribution. The proofs are given in Barmalzan et al. (2017).

Proposition 3.6

Let $Z\sim$EWG($\alpha ,\beta ,p$). Then, (C1) and (C4) hold, for $\alpha \ge 1, \beta \ge 1$ and $p\le (\beta -1)/(\beta +1)$.

Some properties of the Lower-truncated Weibull distribution are discussed in the following proposition. The proofs are obvious.

Proposition 3.7

Let $Z\sim$LTW($\alpha$). Then, the following results are true:

$(\mathrm{i})$ :

(C1) and (C5) hold, for $\alpha \ge 1$, and (C2) and (C6) hold, for $\alpha \le 1$;

$(\mathrm{ii})$ :

(C3) holds, for $\alpha \le 1$ or $\alpha \ge 2$;

$(\mathrm{iii})$ :

(C4) holds, for $\alpha >0$;

$(\mathrm{iv})$ :

$u{r'_Z(u)}/{r_Z(u)}$ is independent of u, for all $\alpha >0$.

Some properties of the generalized gamma distribution are discussed in the following proposition. The proofs of (i)–(iii) are given in Khaledi et al. (2011), whereas (iv) could be found in Ding et al. (2017).

Proposition 3.8

Let $Z\sim$GG(p, q). Then, the following results are true:

$(\mathrm{i})$ :

(C2) holds, for $0<p,q\le 1$, and (C1) holds, for $p,q\ge 1$;

$(\mathrm{ii})$ :

(C4) holds, for all $p>0$ and $q>0$;

$(\mathrm{iii})$ :

(C6) holds, for $p<1$ and $q<1$, and (C5) holds, for $p>1$ and $q>1$.

$(\mathrm{iv})$ :

(C7) holds, for all $0<p\le q$.

The following proposition discusses some properties of the Half-normal distribution. The proof follows from Proposition 3.8 by using the fact that $r_Z(x)=(1/\sqrt{2})r_Y(x/\sqrt{2})$, where Z and Y follow the Half-normal distribution and the generalized gamma distribution (with parameters $p=2$ and $q=1$), respectively.

Proposition 3.9

Let Z have a Half-normal distribution. Then (C1), (C4) and (C5) hold.

Some properties of the Fr$\grave{\mathrm{e}}$chet distribution are discussed in the following proposition. The proofs are omitted.

Proposition 3.10

Let $Z\sim$Fr$\grave{\mathrm{e}}$($\alpha$). Then (C4) and (C7) hold, for all $\alpha >0$.

Some properties of the Pareto distribution are studied in the next proposition. The proofs are omitted.

Proposition 3.11

Let $Z\sim$PD($\alpha$). Then, the following results are true:

$(\mathrm{i})$ :

$u r_Z(u)$ and $u{r'_Z(u)}/{r_Z(u)}$ are independent of u, for all $\alpha >0$;

$(\mathrm{ii})$ :

(C2) and (C3) hold, for all $\alpha >0$;

Remark 3.1

For any random variable Z, the fact ‘$u^2 r'_Z(u)$ is increasing (resp. decreasing) in $u>0$’ equivalently means that (C5) [resp. (C6)] holds.

In the following four remarks, we state some important practical results showing for what baseline distributions (and the corresponding admissible regions of parameters) our results (discussed in Sect. 2) hold. It is worthwhile to mention here that some of the results given in the next remarks have already been stated in the literature, see, for example, Balakrishnan and Zhao (2013a, b), Barmalzan et al. (2017), Bon and Păltănea (2006), Ding et al. (2017), Gupta et al. (2015), Khaledi et al. (2011), Kochar and Torrado (2015), Kochar and Xu (2007), Kundu et al. (2016), Li et al. (2016), Li and Li (2016), Torrado (2015b, 2017), Torrado and Veerman (2012), Zhao and Balakrishnan (2011, 2012), Zhao et al. (2009).

Remark 3.2

It is to be noted that Theorems 2.1–2.5 hold for EW($\alpha ,\beta$) with $\alpha \ge 1$ and $\alpha \beta \ge ~1$, PGW(c, k) with $0<\max \{k,1\}<c$, EWG($\alpha ,\beta ,p$) with $\alpha \ge 1, \beta \ge 1$ and $p\le (\beta -1)/(\beta +1)$, LTW($\alpha$) with $\alpha \ge 1$, GG(p, q) with $p\ge 1$ and $q\ge 1$, and Half-normal distribution as the baseline distributions, whereas the result given in the parenthesis of Theorem 2.4 holds for the baseline distributions FP($\alpha ,\beta ,\gamma$) with $\alpha >0$ and $1<\beta \le \gamma$, GP($\xi$) with $\xi >0$, Burr(c, k) with $0<c\le 1$ and $k>0$, EW($\alpha ,\beta$) with $0<\alpha \le 1$ and $\alpha \beta \le 1$, PGW(c, k) with $0<c\le k$ and $c<1$, LTW($\alpha$) with $0<\alpha \le 1$, GG(p, q) with $0<p,q\le 1$, and PD($\alpha$) with $\alpha >0$ (cf. Propositions 3.1–3.11).

Remark 3.3

Theorems 2.6, 2.7 and 2.10 hold for the baseline distributions EW($\alpha ,\beta$) with $\alpha \ge 2$ and $\beta =1$, PGW(c, k) with $c\ge 2$ and $0<k\le 1$, and LTW($\alpha$) with $\alpha \ge 2$. Further, Theorems 2.8, 2.9(i), (ii) hold for the baseline distributions EW($\alpha ,\beta$) with $\alpha \ge 1$ and $\beta = 1$, PGW(c, k) with $c>k>0$ and $c>1$, LTW($\alpha$) with $\alpha \ge 1$, GG(p, q) with $p> 1$ and $q>1$, Half-normal distribution, and PD($\alpha$) with $\alpha >0$, whereas Theorems 2.9(iii), (iv) and the results given in the parentheses of Theorems 2.9(i), (ii) hold for GP($\xi$) with $\xi >0$, Burr(c, k) with $0<c\le 1$ and $k>0$, EW($\alpha ,\beta$) with $\alpha \le 1$ and $\beta = 1$, PGW(c, k) with $0<c<1$ and $c\le k$, LTW($\alpha$) with $\alpha \le 1$, GG(p, q) with $0<p< 1$ and $0< q<1$, and PD($\alpha$) with $\alpha >0$ as the baseline distributions (cf. Propositions 3.2–3.11, and Remark 3.1).

Remark 3.4

It is to be noted that Theorems 2.11, 2.12, 2.14 and 2.15 hold for EW($\alpha ,\beta$) with $\alpha >0$ and $\beta =1$, PGW(c, k) with $c\ge 3$ and $k\ge 1$, LTW($\alpha$) with $\alpha >0$, and PD($\alpha$) with $\alpha >0$ as the baseline distributions, whereas Theorem 2.13 holds for the baseline distributions FP($\alpha ,\beta ,\gamma$) with $0<\alpha <1\le \beta$ and $\gamma >0$, GP($\xi$) with $\xi >0$, Burr(c, k) with $0<c\le 1$ and $k>0$, EW($\alpha ,\beta$) with $\alpha >0$ and $\beta = 1$, PGW(c, k) with $c>0$ and $k\ge 1$, LTW($\alpha$) with $\alpha >0$, GG(p, q) with $0<p\le q$, Fr$\grave{\mathrm{e}}$($\alpha$) with $\alpha >0$, and PD($\alpha$) with $\alpha >0$ (cf. Propositions 3.1–3.11).

Remark 3.5

Theorems 2.16, 2.17, 2.19 and 2.20 hold for the baseline distributions EW($\alpha ,\beta$) with $\alpha \ge 1$ and $\beta =1$, PGW(c, k) with $c\ge 3$ and $c>k\ge 1$, LTW($\alpha$) with $\alpha \ge 1$, and PD($\alpha$) with $\alpha >0$, whereas Theorem 2.18 holds for EW($\alpha ,\beta$) with $\alpha \ge 1$ and $\beta = 1$, PGW(c, k) with $c>k\ge 1$, LTW($\alpha$) with $\alpha \ge 1$, GG(p, q) with $1<p\le q$, and PD($\alpha$) with $\alpha >0$ as the baseline distributions (cf. Propositions 3.4–3.11).

4 Concluding remarks

This paper develops a theory of stochastic comparisons for two minimum order statistics that are formed from two different sets of random variables having different/the same location as well as different/the same scale parameters. The obtained general results hold also for many well known specific baseline (for location–scale family) distributions, namely, Feller–Pareto distribution, generalized Pareto distribution, Burr distribution, exponentiated Weibull distribution, Power generalized Weibull distribution, generalized gamma distribution, Half-normal distribution, Fr$\grave{\mathrm{e}}$chet distribution and Pareto distribution. We also have obtained the corresponding ranges for parameters for these specific distributions, where the comparisons hold. We believe that our study is meaningful both from theoretical and practical points of view.

It is well known that series systems in reliability can be equivalently described via the minimum order statistics. Thus our study answers an important question: what relations should exist among parameters of the sets of the corresponding random variables in order the one series system to dominate the other with respect to different stochastic orders. Due to analytical complexity and substantial differences in the corresponding proofs and reasoning, it is unrealistic to combine the study of stochastic comparisons for minimum and maximum order statistics in a single paper. Therefore, the latter case has been reported separately (Hazra et al. 2017). Furthermore, as series and parallel systems are the building blocks for general coherent systems, our study can be considered as the first step in the direction of developing a theory of stochastic dominance for general coherent systems.

A real application of the proposed results could be as follows. Let $X_{1:n}$ be the minimum accident-free distance for one type of an explosive that may explode any time during its transportation and let $Y_{1:n}$ be that for another type of an explosive. The possibility of an explosion depends on certain parameters viz. nature of an explosive, condition of the road etc. All these indexes may be represented by different parameters $\lambda _i$ and $\mu _i$ for calculating the accident-free distance of transportation. Our results state that if one set of parameters is dominated by another set, then the minimum accident-free distance for transporting one type of an explosive will dominate that of the other one.

We conclude our discussion by mentioning the fact that the straightforward corollaries corresponding to each theorem (discussed in this paper) could be formulated similar to the one given below for Theorem 2.10.

Corollary 4.1

Let $X_i\sim$ LS$(\lambda _i,\theta _i)$ and $Y_i\sim$ LS$(\lambda ,\theta )$, for $i=1,2,\ldots ,n$. Suppose that the set of conditions $\{{\varvec{\lambda }}\in \mathcal {E},\; {\varvec{\theta }}\in \mathcal {E}\}$ or $\{{\varvec{\lambda }}\in \mathcal {D},\;{\varvec{\theta }}\in \mathcal {D}\}$ holds. If (C1) and (C3) hold then

$$\begin{aligned} \lambda \ge \frac{1}{n}\sum \limits _{i=1}^n \lambda _i\text { and } \frac{1}{\theta }\le \frac{1}{n}\sum \limits _{i}^n \frac{1}{\theta _i}\;\implies \; X_{1:n}\le _{hr} Y_{1:n}. \end{aligned}$$

Appendix

Proof of Theorem 2.1

We only prove the result under the assumption $\{{\varvec{\lambda }}\in \mathcal {E},\; {\varvec{\mu }}\in \mathcal {E},\;{\varvec{\theta }}\in \mathcal {E}\}$. It could be proven in the same line under the second assumption. Note that

$$\begin{aligned} \bar{F}_{X_{1:n}}(x;{{\varvec{\lambda }}},{{\varvec{\theta }}})\;\text { is increasing in each }\lambda _i,\;\text {for }i=1,2,\ldots ,n. \end{aligned}$$

(4.1)

Further,

$$\begin{aligned} \frac{\partial {\bar{F}}_{X_{1:n}}(x;{{\varvec{\lambda }}},{{\varvec{\theta }}})}{\partial \lambda _k}\text { is decreasing in }k=1,2,\ldots ,n, \end{aligned}$$

if, for $1\le p\le q\le n$,

$$\begin{aligned} \frac{1}{\theta _p}r\left( \frac{x-\lambda _p}{\theta _p}\right) \ge \frac{1}{\theta _q}r\left( \frac{x-\lambda _q}{\theta _q}\right) . \end{aligned}$$

(4.2)

Consider the following two cases.

• Case I Let $x\le \lambda _p\le \lambda _q$ or $\lambda _p\le x\le \lambda _q$. Then, (4.2) is trivially true.

• Case II Let $\lambda _p\le \lambda _q\le x$. Then, (4.2) holds because $r(\cdot )$ is an increasing function.

Thus, (4.2) holds for both the cases, and hence, by Lemma 3.3 of Kundu et al. (2016), we get that

$$\begin{aligned} {\bar{F}}_{X_{1:n}}(x;{{\varvec{\lambda }}},{{\varvec{\theta }}})\text { is Schur-concave in } {{\varvec{\lambda }}}\in \mathcal {E}. \end{aligned}$$

(4.3)

Finally, on using (4.1) and (4.3) in Lemma 4.3 of Kundu et al. (2016), we get the required result. $\square$

Proof of Theorem 2.6

We only prove the result under the assumption $\{{\varvec{\lambda }}\in \mathcal {E},\; {\varvec{\mu }}\in \mathcal {E},\;{\varvec{\theta }}\in \mathcal {E}\}$. It could be proven in the same line under the second assumption. Note that

$$\begin{aligned} r_{X_{1:n}}(x;{{\varvec{\lambda }}},{{\varvec{\theta }}})\;\text { is decreasing in each }\lambda _i,\;\text {for }i=1,2,\ldots ,n. \end{aligned}$$

(4.4)

Further,

$$\begin{aligned} \frac{\partial r_{X_{1:n}}(x;{{\varvec{\lambda }}},{{\varvec{\theta }}})}{\partial \lambda _k}\text { is increasing in }k=1,2,\ldots ,n, \end{aligned}$$

if, for $1\le p\le q\le n$,

$$\begin{aligned} \frac{1}{\theta ^2_p} r'\left( \frac{x-\lambda _p}{\theta _p}\right) \ge \;\frac{1}{\theta _q^2} r'\left( \frac{x-\lambda _q}{\theta _q}\right) . \end{aligned}$$

(4.5)

Since, $\lambda _q\ge \lambda _p$ and $\theta _q\ge \theta _p$, we have that

$$\begin{aligned} \frac{1}{\theta _p^2}\ge \frac{1}{\theta _q^2}\; \text{ and } \;\frac{x-\lambda _p}{\theta _p}\ge \frac{x-\lambda _q}{\theta _q}. \end{aligned}$$

(4.6)

Consider the following three cases:

• Case I Let $\lambda _q\ge \lambda _p\ge x$. Then (4.5) holds trivially.

• Case II Let $\lambda _q\ge x\ge \lambda _p$. Then (4.5) follows from (4.6) and the fact that $r(\cdot )$ is an increasing function.

• Case III Let $x\ge \lambda _q\ge \lambda _p$. Then (4.5) follows from (4.6) and the fact that r(u) is both increasing and convex in u. Thus, for all the cases, (4.5) holds. So, by Lemma 3.3 of Kundu et al. (2016), we have that

  $$\begin{aligned} r_{X_{1:n}}(x;{{\varvec{\lambda }}},{{\varvec{\theta }}})\text { is Schur-convex in } {{\varvec{\lambda }}}\in \mathcal {E}. \end{aligned}$$

  (4.7)

Finally, on using (4.4) and (4.7) in Lemma 4.3 of Kundu et al. (2016), we get the required result. $\square$

Proof of Theorem 2.11

Let

$$\begin{aligned}&\mathcal {F}=\{{\varvec{\mu }}=(\mu _1,\mu _2,\ldots ,\mu _n):\;\mu _i=\theta _1^{-1},\\&\qquad \text { for }1\le i\le n_1 \text { and } \mu _i=\theta _2^{-1}, \text { for }n_1+1\le i\le n \}. \end{aligned}$$

Further, let $u(x)=xr(x)$, $v(x)=xr'(x)/ r(x)$, and $\omega (x)=u(x)v'(x)$, for all $x>0$. Note that $r_{X_{1:n}}(x)/r_{Y_{1:n}}(x)$ is decreasing in x if

$$\begin{aligned} \Lambda (\mu _1,\mu _2,\ldots ,\mu _n){\mathop {=}\limits ^\mathrm{def.}}\frac{\sum \limits _{i=1}^n u(\mu _i (x-\lambda ))v(\mu _i (x-\lambda ))}{\sum \limits _{i=1}^n u(\mu _i (x-\lambda ))} \end{aligned}$$

is Schur-concave in $(\mu _1,\mu _2,\ldots ,\mu _n)\in \mathcal {F}$. Consider the following two cases:

• Case I Let $1\le i,j\le n_1$ or $n_1+1\le i,j\le n$. Then

  $$\begin{aligned} \frac{\partial \Lambda }{\partial \mu _i}- \frac{\partial \Lambda }{\partial \mu _j}=0 \end{aligned}$$

• Case II Let $1\le i\le n_1$ and $n_1+1\le j\le n$. Then

  $$\begin{aligned}&\frac{\partial \Lambda }{\partial \mu _i}- \frac{\partial \Lambda }{\partial \mu _j}{\mathop {=}\limits ^\mathrm{sgn}}n_1n_2\{v(a_1 (x-\lambda ))-v(a_2 (x-\lambda ))\}\{ u'(a_1 (x-\lambda ))u(a_2 (x-\lambda ))\\&\qquad +\, u'(a_2 (x-\lambda ))u(a_1 (x-\lambda ))\}\\&\qquad +\, \{n_1 u(a_1 (x-\lambda ))+n_2 u(a_2 (x-\lambda ))\}\{n_1 \omega (a_1 (x-\lambda ))- n_2\omega (a_2 (x-\lambda ))\}, \end{aligned}$$

  where $a_1=\theta _1^{-1}$ and $a_2=\theta _2^{-1}$. From the hypothesis, we have that u(x) is increasing in x, and v(x) is decreasing and concave in x. This implies that w(x) is non-positive and decreasing in x. Thus, it follows that $\frac{\partial \Lambda }{\partial \mu _i}-\frac{\partial \Lambda }{\partial \mu _j}\le 0$ whenever the set of conditions $\{(\theta _1,\theta _2)\in ~\mathcal {E}, (\delta _1,\delta _2)\in \mathcal {E},(n_1,n_2)\in \mathcal {D}\}$ holds. Further, $\frac{\partial \Lambda }{\partial \mu _i}-\frac{\partial \Lambda }{\partial \mu _j}\ge 0$ whenever the set of conditions $\{(\theta _1,\theta _2)\in \mathcal {D}, (\delta _1,\delta _2)\in \mathcal {D}, (n_1,n_2)\in \mathcal {E}\}$ holds. Hence the result follows from Lemma 3.3 of Kundu et al. (2016) and Theorem A.3 of Marshall et al. (2011, p. 83).$\square$

Proof of Theorem 2.14

Suppose that the first set of conditions holds. Then, the weak submajorization order gives that $\theta _2^{-1}\ge \delta _2^{-1}$ and $b\theta _1^{-1}+n_2\theta _2^{-1}\ge b\delta _1^{-1}+n_2\delta _2^{-1}$, for $b=n_1, n_1-1,\ldots , 2, 1$. If $n_1\theta _1^{-1}+n_2\theta _2^{-1}= n_1\delta _1^{-1}+n_2\delta _2^{-1}$ holds, then the result follows from Theorem 2.11. Suppose that $n_1\theta _1^{-1}+n_2\theta _2^{-1}> n_1\delta _1^{-1}+n_2\delta _2^{-1}$. Then, there exists some $\theta$ such that $n_1\theta _1^{-1}+n_2\theta ^{-1}= n_1\delta _1^{-1}+n_2\delta _2^{-1}$ and $\theta _2^{-1} >\theta ^{-1} \ge \delta _2^{-1}$. Let $X^*_{1:n}$ be the minimum order statistic formed from the set of random variables ($X_1^*,X_2^*,\ldots ,X_n^*$), where $X_i^*\sim LS(\lambda ,\theta _1)$, for $i=1,2,\ldots ,n_1$ and $X_j^*\sim LS(\lambda ,\theta )$, for $j=n_1+1,n_1+2,\ldots , n_1+n_2\;(=n)$. Then, on using Lemma 4.4 of Kundu et al. (2016), $X^*_{1:n}\ge _{R-HR}Y_{1:n}$ follows from Theorem 2.11. Further, note that $\theta _2^{-1} >\theta ^{-1} \ge \theta _1^{-1}$ and

$$\begin{aligned}&\left( \underbrace{{\theta _1}^{-1},{\theta _1}^{-1},\ldots ,{\theta _1}^{-1},}_{n_1} \underbrace{{\theta _2}^{-1},{\theta _2}^{-1},\ldots ,{\theta _2}^{-1}}_{n_2}\right) \\&\quad \succeq _{w}\left( \underbrace{{\theta _1}^{-1},{\theta _1}^{-1},\ldots ,{\theta _1}^{-1},}_{n_1} \underbrace{{\theta }^{-1},{\theta }^{-1},\ldots ,{\theta }^{-1}}_{n_2}\right) . \end{aligned}$$

Thus, $X_{1:n}\ge _{R-HR}X^*_{1:n}$ follows from Theorem 2.13. Hence $X_{1:n}\ge _{R-HR}Y_{1:n}$. The proof follows in a similar way under the second set of conditions.$\square$

Proof of Theorem 2.15

Writing $D_2(x)= r_{X_{1:n}}(x;\lambda ,{{\varvec{\theta }}})/ r_{Y_{1:n}}(x;\lambda ,\theta )$, we have

where the first inequality follows from (C4), (C7) and equation (1.5) of Mitrinović et al. (1993, p. 240) whereas the second inequality follows from (C7) and (C8). $\square$

Proof of Lemma 3.1(i)

Writing $\Delta _1(u)=ur'_Z(u)/r_Z(u)$, we have

$$\begin{aligned} \Delta _1(u)=-\left( 1+\frac{\alpha }{\gamma }\right) +\left( \frac{\alpha +\beta }{\gamma }\right) \left( \frac{1}{1+u^{\frac{1}{\gamma }}}\right) +ur_Z(u). \end{aligned}$$

(4.8)

Differentiating the above expression with respect to u, we get

$$\begin{aligned} \Delta _1'(u)=\frac{1}{u}\left[ -\left( \frac{\alpha +\beta }{\gamma ^2}\right) \left\{ \frac{u^{\frac{1}{\gamma }}}{\left( 1+u^{\frac{1}{\gamma }}\right) ^2}\right\} +ur_Z(u)\left( 1+\Delta _1(u)\right) \right] . \end{aligned}$$

(4.9)

From Lemma 3.1, we have that $\lim _{u \rightarrow 0}\Delta _1(u)=-\,1+\beta /\gamma$ and $\lim _{u \rightarrow \infty } \Delta _1(u)=-\,1$. If possible, let $\min _{u>0}\Delta _1(u)<-\,1$. Then, $\Delta _1(u)$ has to cross the level $u=-\,1$ at least once. Let $v\in (0,\infty )$ be any point such that $\Delta _1(v)=-\,1$. Then, from (4.9), we have that $\Delta _1'(v)<0.$ This means that, if $\Delta _1(u)$ crosses the level $u=-\,1$, then it will cross exactly once and from above. Assume that $u_0$ ($>0$) is the point where $\Delta _1(u)$ crosses the level $u=-\,1$. Then, for $u_0<u< \infty$, we have $\Delta _1(u)<-\,1$. On using this in (4.9), we get that $\Delta _1(u)$ is decreasing in $u\in (u_0,\infty )$. This gives that $\Delta _1(u)> \lim u_{u\rightarrow \infty }\Delta _1(u)=-\,1$, which is a contradiction to the fact that $\Delta _1(u)<-\,1$. Thus, the point $u_0$ does not exist, and hence the minimum of $\Delta _1(u)$ is always larger or equal to $-\,1$. Therefore, the result is proved.$\square$

Proof of Lemma 3.1(ii)

If possible, let $\max _{u>0}\Delta _1(u)>-\,1+\beta /\gamma$. Then, $\Delta _1(u)$ has to cross the level $u=-\,1+\beta /\gamma$ at least once and from above because $\lim _{u \rightarrow 0}\Delta _1(u)=-\,1+\beta /\gamma$ and $\lim _{u \rightarrow \infty } \Delta _1(u)=-\,1$. Let $w\in (0,\infty )$ be any point such that $\Delta _1(w)=-\,1+\beta /\gamma$. Then, from (4.8), we have

$$\begin{aligned} wr_Z(w)=\left( \frac{\alpha +\beta }{\gamma }\right) \left( \frac{w^{\frac{1}{\gamma }}}{1+w^{\frac{1}{\gamma }}}\right) . \end{aligned}$$

On using this and the fact that $g(u){\mathop {=}\limits ^{\text {def.}}}\beta -1/(1+u^{1/\gamma })$ is positive for all $u>0$, we have, from (4.9), that $\Delta _1'(w)>0$. This means that, if $\Delta _1(u)$ crosses the level $u=-\,1+\beta /\gamma$, then it will cross exactly once and from below. In other wards, $\Delta _1(u)$ never crosses the level $u=-\,1+\beta /\gamma$ from above, which is a contradiction. Thus, the maximum of $\Delta _1(u)$ does not exceed the level $u=-\,1+\beta /\gamma$. Hence, the result follows.$\square$

Proof of Proposition 3.1(iii)

From Proposition 3.1(i) and Lemma 3.1, we have that $ur_Z(u)$ is increasing in u with $\lim _{u \rightarrow 0}ur_Z(u)=0$ and $\lim _{u \rightarrow \infty }u r_Z(u)=\alpha /\gamma$. Thus,

$$\begin{aligned} ur_Z(u)\le \frac{\alpha }{\gamma },\quad \text { for all }u>0. \end{aligned}$$

On using this and (4.9), we have

$$\begin{aligned} \left[ u\frac{\tilde{r}_Z'(u)}{\tilde{r}_Z(u)}\right] '\le & {} \left( \frac{\alpha +\beta }{\gamma }\right) \left( \frac{u^{-1}}{1+u^{\frac{1}{\gamma }}}\right) \xi _1(u), \end{aligned}$$

(4.10)

where

$$\begin{aligned} \xi _1(u)=u r_Z(u)-\frac{u^{\frac{1}{\gamma }}}{\gamma \left( 1+u^{\frac{1}{\gamma }}\right) }. \end{aligned}$$

Thus, to prove the result, it suffices to show that $\xi _1(u)\le 0$. Now, on using Lemma 3.1, we have

$$\begin{aligned} \lim _{u\rightarrow 0} \xi _1(u)=0 \text { and } \lim _{u\rightarrow \infty } \xi _1(u)=\frac{\alpha -1}{\gamma }<0. \end{aligned}$$

Let us assume that $\max _{0<u<\infty }\xi _1(u)>0$. Then, $\xi _1(u)$ has to cross the level zero at least once and from above. This means that there must exist a point $w\in (0,\infty )$ such that $\xi _1(w)=0$ and $\xi _1'(w)<0$. Let $v\in (0,\infty )$ be any point such that $\xi _1(v)=0$. On using this, we get, after some algebraic calculation, that $\xi _1'(v)>0$. This shows that there does not exist any such point w where both $\xi _1(w)=0$ and $\xi _1'(w)<0$ hold. Thus, our assumption is not correct, and hence $\xi _1(u)\le 0$, for all $u> 0$. Therefore, the result is proved. $\square$