Mean residual life order among largest order statistics arising from resilience-scale models with reduced scale parameters

Abstract

In this paper, we identify some conditions to compare the largest order statistics from resilience-scale models with reduced scale parameters in the sense of mean residual life order. As an example of the established result, the exponentiated generalized gamma distribution is examined. Also, for the special case of the scale model, power-generalized Weibull and half-normal distributions are investigated.

1. Introduction

If operating of a system comprising $n$ components depends to at least $k$ active components, then it is called a $k$-out-of-$n$ system. The lifetimes of $k$-out-of-$n$ systems thus can be described by order statistics arising from the lifetimes of their components. Due to this intimate relation, order statistics play a fundamental rule in the context of reliability theory. In this regard, extreme order statistic, among others, corresponding to the lifetimes of series and parallel systems have received more attentions because of their importance in analyzing of the lifetimes of complex systems; see Barlow and Proschan [7]. For elaborate discussions on order statistics and their applications, one may refer to Balakrishnan and Rao [4,5] and David and Nagaraja [11].

Consider a nonnegative random variable $T$ with distribution function $F_{T}$ and assume that $F_{T}(x)=G^{\theta }(\delta x)$ for all $x\in \mathbb {R}_+(=[0,\infty ))$ wherein $G$ is an absolutely continuous distribution function (centered on $\mathbb {R}_+$) with corresponding reversed hazard rate function $\tilde {r}$. Then, it is said that $T$ follows the resilience-scale (RS) model with baseline distribution $G,$ resilience parameter $\theta \in \mathbb {R}^+$ and scale parameter $\delta \in \mathbb {R}^+,$ written as $T\sim RS(G;\theta,\delta )$. Note that the RS model becomes the scale model when $\theta =1$. If $\tilde {r}_{T}$ denotes the reversed hazard rate function of $T,$ then one can see that $r_{T}(x)=\theta \delta \tilde {r}(\delta x)$ for all $x\in \mathbb {R}_+$. Therefore, the RS model can be viewed as the scaled version of the proportional hazard rate model. On the other hand, the RS model can be achieved by using the exponentiation method on the scale model, which for this reason, it is also called as the exponentition scale model in the literature. For a comprehensive discussion on the exponentiation method and its applications, we refer the readers to AL-Hussaini and Ahsanullah [1]. It should be mentioned that the RS model contains some well-known life distributions such as generalized exponential distribution, exponentiated Weibull distribution, exponentiated Lomax distribution, generalized Rayleigh distribution, exponentiated gamma distribution and exponentiated generalized gamma distribution.

In the recent years, some studies have been focused on stochastic comparisons between the largest order statistics for several specific cases of the RS model. For example, one can see Balakrishnan et al. [3] and Kundu et al. [21] for the case of generalized exponential distribution; Fang and Zhang [14], Kundu and Chowdhury [20] and Barmalzan et al. [8] for the case of exponentiated Weibull distribution; Fang and Xu [13] for the case of exponentiated gamma distribution; Haidari et al. [16] and Haidari and Payandeh Najafabadi [15] for the case of the exponentiated generalized gamma distribution. However, stochastic ordering relations between the largest order statistics based on random variables following the general RS models have not received the attentions they deserve. Indeed, to the best of our knowledge, only few works in this direction are published so far including Zhang et al. [31], Haidari et al. [17] and Lu et al. [23].

Mean residual life function is a key tool in reliability, life testing and survival analysis. Provided that an item is of age $t\in \mathbb {R}_+$, the remaining lifetime after $t$ is a random variable whose expected value is called the mean residual life (MRL) function at time $t$. As one can see, the MRL function sums up the entire residual life distribution of an item while the failure (hazard) rate function describes the effect of an immediate failure. From this point of view, the MRL function is likely to be more efficient than the failure (hazard) rate function. The MRL function has wide applications in other areas such as renewal theory, demography, social sciences and actuarial sciences; see Chapter 4 of Lai and Xie [22] and the references therein.

Before going into the background of the idea investigated in this paper, let us briefly recall some definitions. For two nonnegative random variables $T_1$ and $T_2$ with survival functions $\bar {F}_{T_1}$ and $\bar {F}_{T_2}$, and mean residual functions $m_{T_1}(t)=[\bar {F}_{T_1}(t)]^{-1}\int _{t}^{\infty }\bar {F}_{T_1}(u)\,du$ and $m_{T_2}(t)=[\bar {F}_{T_2}(t)]^{-1}\int _{t}^{\infty }\bar {F}_{T_2}(u)\,du$, it is said that $T_1$ is larger than $T_2$ with respect to the mean residual life order, denoted by $T_1\ge _{{\rm mrl}}T_2$, if $m_{T_1}(t)\ge m_{T_2}(t)$ for all $t\in \mathbb {R}_+;$ see Chapter 2 of Shaked and Shanthikumar [27] for more details on the mean residual life order and its properties. Let $u_{1:n}\le \dots \le u_{n:n}$ and $v_{1:n}\le \dots \le v_{n:n}$ denote the increasing arrangements of the components of nonnegative vectors $\boldsymbol {u}=(u_1,\ldots,u_n)$ and $\boldsymbol {v}=(v_1,\ldots,v_n)$, respectively. Then, $\boldsymbol {u}$ is said to reciprocal majorize $\boldsymbol {v}$, written as $\boldsymbol {u}\stackrel {{\rm rm}}{\succ }\boldsymbol {v}$, if $\sum _{i=1}^{k}u^{-1}_{i:n}\ge \sum _{i=1}^{k}v^{-1}_{i:n}$ for all $k=1,\ldots,n$. For additional details on the reciprocal mjorization order and its application, one may refer to Zhao and Balakrishnan [32].

Suppose $T_1,\ldots,T_n$ and $T^{*}_1,\ldots,T^{*}_n$ are two sets of independent exponential random variables with $T_i$, $T_j$, $T^{*}_i$ and $T^{*}_j$ having the respective hazard rates $\delta _1$, $\delta _2$, $\delta ^{*}_1$ and $\delta ^{*}_2$ for $i=1,\ldots,l$ and $j=l+1,\ldots,n$ ($1\le l\le n-1$), and let $T_{n:n}$ (resp. $T^{*}_{n:n}$) denotes the largest order statistic based on $T_1,\ldots,T_n$ (resp. $T^{*}_1,\ldots,T^{*}_n$ ). In the sequel, a vector with all its elements being one is represented by $\boldsymbol {1}_{k}$. In Open Problem 2 of Balakrishnan and Zhao [6], the following idea concerning the mean residual life order between $T_{n:n}$ and $T^{*}_{n:n}$ is proposed:

• Under the assumptions $\delta _1\le \delta ^{*}_1\le \delta ^{*}_2\le \delta _2$ and $(\delta _1\boldsymbol {1}_{l},\delta _2\boldsymbol {1}_{n-l})\stackrel {{\rm rm}}{\succ }(\delta ^{*}_1\boldsymbol {1}_{l},\delta ^{*}_2\boldsymbol {1}_{n-l})$, does the ordering $T_{n:n}\ge _{{\rm mrl}}T^{*}_{n:n}$ hold?

For $\delta _1\le \delta _2$, set

$$\Omega(\delta_1,\delta_2)=\{(x,y)\in{\mathbb{R}^+}^{2}: \delta_1\le x\le y\le\delta_2\ \text{and}\ l\delta^{{-}1}_1+(n-l)\delta^{{-}1}_2\ge l x^{{-}1}+(n-l) y^{{-}1}\}.$$

Now, the foresaid idea can be restated as follows:

• If $(\delta ^{*}_1,\delta ^{*}_2)\in \Omega (\delta _1,\delta _2)$, then does the ordering $T_{n:n}\ge _{{\rm mrl}}T^{*}_{n:n}$ hold?

Zhao and Balakrishnan [33] showed that the answer of the above questions is positive for the special case when $n=2$. However, the complete answer is provided by Wang and Cheng [29] with the aid of an effective method which is new in the context of stochastic orderings of order statistics; see Wang [28] and Wang and Cheng [30] for further details on this method and its applications.

The idea investigated in this paper is concerning the mean residual life order between the largest order statistics in the RS models. Consider two nonnegative random vectors $(T_1,\ldots,T_n)$ and $(T^{*}_1,\ldots,T^{*}_n)$ with $T_i\sim RS(G;\theta _i,\delta _1)$, $T_j\sim RS(G;\theta _j,\delta _2)$, $T^{*}_i\sim RS(G;\theta _i,\delta ^{*}_1)$ and $T^{*}_j\sim RS(G;\theta _j,\delta ^{*}_2)$ for $i=1,\ldots,l$ and $j=l+1,\ldots,n$. Assume that the region $\Delta _{\theta _1,\theta _2}(\delta _1,\delta _2)$ is formed by the lines $y=x$ and $y=\delta _2$, and the curve $\theta _1 x^{-1}+\theta _2 y^{-1}=\theta _1\delta ^{-1}_1+\theta _2\delta ^{-1}_2$ for $\delta _1\le \delta _2$ and $\theta _i\ge 1$, $i=1,2$. The graph of this region is plotted in Figure 1 in which $\delta _{H}=(\theta _1+\theta _2)/(\theta _1\delta ^{-1}_1+\theta _2\delta ^{-1}_2)$ is the weighted harmonic mean of $\delta _1$ and $\delta _2$ with corresponding weights $\theta _1$ and $\theta _2$. Set $\xi _1=\sum _{i=1}^{l}\theta _i$ and $\xi _{2}=\sum _{j=l+1}^{n}\theta _j$. We will find some conditions on the baseline distribution $G$ such that, for $(\delta ^{*}_1,\delta ^{*}_2)\in \Delta _{\xi _1,\xi _2}(\delta _1,\delta _2)$, the ordering $T_{n:n}\ge _{{\rm mrl}}T^{*}_{n:n}$ holds. We will also examine this result when the baseline distribution is the generalized gamma. When $\theta _i=1$ for all $i=1,\ldots,n$ (the scale model with reduced heterogeneity of scale parameters), the power-generalized Weibull and half-normal distributions are investigated as the examples. For many well-known distributions, a huge body of literature exists concerning comparisons of their largest order statistics with respect to magnitude orders such as the usual stochastic, hazard rate, and likelihood ratio orders. But, for the mean residual life order, attentions has been focused just on the exponential distribution whereas other distributions remain noticeably absent in the literature. The results established in this paper fill this gape by extension of the previous ones from the exponential framework to the exponentiated generalized gamma, power-generalized Weibull and half-normal frameworks.

Figure 1. Plot of the region $\Delta _{\theta _1,\theta _2}(\delta _1,\delta _2)$.

The general structure of the paper can be summarized as follows: In Section 2, we present the main results. Several examples and illustrations are stated in Section 3. Finally, some discussions are made in Section 4. Throughout the paper, we write $P\stackrel {{\rm sgn}}=Q$ to mean that $P$ and $Q$ have the same sign. Furthermore, for any differentiable function $w(x)$, $w'(x)$ denotes the first derivative of $w(x)$ with respect to $x$ while the notion $\partial _{i}b(x_1,x_2)$ is used for the partial derivative of any differentiable function $b(x_1,x_2)$ with respect to $x_i$, $i=1,2$.

2. Main results

Here, we compare the largest order statistics arising from independent random variables following heterogeneous RS models with respect to the mean residual life order. In what follows, everywhere we use the notions $(T_1,\ldots,T_n)\sim RS(G;\boldsymbol {\theta },\boldsymbol {\delta }_{l})$ and $(T^{*}_1,\ldots,T^{*}_n)\sim RS(G;\boldsymbol {\theta },\boldsymbol {\delta }^{*}_{l})$ wherein $\boldsymbol {\theta }=(\theta _1,\ldots,\theta _n)$, $\boldsymbol {\delta }_l=(\delta _1\boldsymbol {1}_{l},\delta _2\boldsymbol {1}_{n-l})$ and $\boldsymbol {\delta }^{*}_l=(\delta ^{*}_1\boldsymbol {1}_{l},\delta ^{*}_2\boldsymbol {1}_{n-l})$, it means that $T_i\sim RS(G;\theta _i,\delta _1)$, $T_j\sim RS(G;\theta _j,\delta _2)$, $T^{*}_i\sim RS(G;\theta _i,\delta ^{*}_1)$ and $T^{*}_j\sim RS(G;\theta _j,\delta ^{*}_2)$ for $i=1,\ldots,l$ and $j=l+1,\ldots,n$. Furthermore, the survival, density, hazard rate and reversed hazard rate functions of the baseline distribution function $G$, centered on $\mathbb {R}_+$, are respectively denoted by $\bar {G}$, $g$, $r=g/\bar {G}$ and $\tilde {r}=g/G$. Set $\alpha (x)=x\tilde {r}(x)$, $\gamma (x)=x\tilde {r}'(x)/\tilde {r}(x)$, $F(x;u_1,u_2)=[G(u_1x)]^{\theta _1}[G(u_2x)]^{\theta _2}$ and $\bar {F}(x;u_1,u_2)=1-F(x;u_1,u_2)$ for all $x\in {\mathbb {R}}^+$ and $(u_1,u_2)\in {\mathbb {R}^+}^2$. Under this setting, it can be easily seen that

(1) \begin{align} \gamma(x)& =\frac{xg'(x)}{g(x)}-\alpha(x), \quad x\in \mathbb{R}^+, \end{align}

(2) \begin{align} \frac{xr'(x)}{r(x)}& =\frac{xg'(x)}{g(x)}+xr(x), \quad x\in \mathbb{R}^+. \end{align}

To prove the main result, we need a series of lemmas which are presented in the sequel.

Lemma 1. (Mitrinović et al. [24, p. 71])

Let $y_k\in (0,1)$ and $\nu _k \geq 1$ for all $k=1,\ldots, n$. Then, we have

$$1-\prod_{k=1}^{n} (1-y_k)^{\nu_k} \leq \sum _{k=1}^{n} \nu_k y_k.$$

Lemma 2. (Mitrinović et al. [24, p. 340])

Consider three vectors $\pmb {c}=(c_1,\ldots,c_n)\in \mathbb {R}^n$, $\pmb {q}=(q_1,\ldots,q_n)\in \mathbb {R}^n$ and $\pmb {d}=(d_1,\ldots,d_n)\in \mathbb {R}^{+n}$. Then, we have

$$\min \left\{\frac{c_1}{d_1},\ldots, \frac{c_n}{d_n}\right\} \leq \frac{\sum_{i=1}^n q_ic_i}{\sum_{i=1}^n q_id_i}\leq \max \left\{ \frac{c_1}{d_1},\ldots , \frac{c_n}{d_n}\right\}.$$

Lemma 3. Suppose that $\theta _i\ge 1$ for $i=1,2$. Let the function $L(\cdot ;t):\mathbb {R}^+\to \mathbb {R}^+$ be defined as:

$$L(x;t)=\frac{\bar{F}(x;1,t)}{F(x;1,t)\alpha(x)},$$

wherein $t\ge 1$. Assume that the following conditions hold:

1. (a₁) $r(x)$ is increasing in $x\in \mathbb {R}^+;$

2. (a₂) $\alpha (x)/\bar {G}(x)$ is increasing in $x\in \mathbb {R}^+$.

Then, $L(x;t)$ is decreasing in $x\in \mathbb {R}^+$.

Proof. Since $F'(x;1,t)=(\theta _1 \tilde {r}(x)+\theta _2t\tilde {r}(tx))F(x;1,t)$ for all $x\in \mathbb {R}^+$, then we find

(3) \begin{align} (xL(x;t))'& \stackrel{{\rm sgn}}={-}F'(x;1,t)F(x;1,t)\tilde{r}(x)-(F'(x;1,t)\tilde{r}(x)+F(x;1,t)\tilde{r}'(x))\bar{F}(x;1,t)\nonumber\\ & ={-}F'(x;1,t)F(x;1,t)\tilde{r}(x)-F'(x;1,t)\bar{F}(x;1,t)\tilde{r}(x)-F(x;1,t)\bar{F}(x;1,t)\tilde{r}'(x)\nonumber\\ & ={-}F'(x;1,t)\tilde{r}(x)(F(x;1,t)+\bar{F}(x;1,t))-F(x;1,t)\bar{F}(x;1,t)\tilde{r}'(x)\nonumber\\ & ={-}F'(x;1,t)\tilde{r}(x)-F(x;1,t)\bar{F}(x;1,t)\tilde{r}'(x)\nonumber\\ & ={-}(\theta_1\tilde{r}(x)+\theta_2t\tilde{r}(tx))F(x;1,t)\tilde{r}(x)-F(x;1,t)\bar{F}(x;1,t)\tilde{r}'(x)\nonumber\\ & ={-}\frac{F(x;1,t)\bar{F}(x;1,t)\tilde{r}(x)}{x}\left(\frac{\theta_1x\tilde{r}(x)+\theta_2tx\tilde{r}(tx)}{\bar{F}(x;1,t)}+\gamma(x)\right)\nonumber\\ & \stackrel{{\rm sgn}}={-}\left(\frac{\theta_1\alpha(x)+\theta_2\alpha(tx)}{\bar{F}(x;1,t)}+\gamma(x)\right),\quad x\in\mathbb{R}^+. \end{align}

We also have

(4) \begin{align} \frac{\theta_1\alpha(x)+\theta_2\alpha(tx)}{\bar{F}(x;1,t)}& \ge \frac{\theta_1\alpha(x)+\theta_2\alpha(tx)}{\theta_1 \bar{G}(x)+\theta_2 \bar{G}(tx)}\nonumber\\ & \ge \min \left\{\frac{\alpha(x)}{\bar{G}(x)},\frac{\alpha(tx)}{\bar{G}(tx)}\right\}\nonumber\\ & =\frac{\alpha(x)}{\bar{G}(x)},\quad x\in\mathbb{R}^+, \end{align}

wherein the first inequality is obtained from Lemma 1, the second inequality is derived from Lemma 2 and finally the last identity is established based on Condition ($a_2$) and the restriction $t\ge 1$. Now, upon combining Eqs. (1), (2) and (4), it follows from Condition ($a_1$) that

\begin{align*} \frac{\theta_1\alpha(x)+\theta_2\alpha(tx)}{\bar{F}(x;1,t)}+\gamma(x)& \ge \frac{\alpha(x)}{\bar{G}(x)}+\frac{xg'(x)}{g(x)}-\alpha(x)\\ & =\frac{xg'(x)}{g(x)}+xr(x)\\ & =\frac{xr'(x)}{r(x)}\\ & \geq 0,\quad x\in\mathbb{R}^+, \end{align*}

which confirms the right-hand side of (3) is non-positive. Thus, $xL(x;t)$ is decreasing in $x\in \mathbb {R}^+$ and so, $L(x;t)$ is also decreasing in $x\in \mathbb {R}^+$, as required.

Lemma 4. For $t\in \mathbb {R}^+$ and $u_2\ge u_1$, let the function $\Xi (\cdot ;t):\mathbb {R}^+\rightarrow \mathbb {R}$ be defined as:

$$\Xi(x;t)=\alpha(t)F(1;u_1,u_2)\bar{F}\left(\frac{x}{t};u_1,u_2\right) -\alpha(x)F\left(\frac{x}{t};u_1,u_2\right)\bar{F}(1;u_1,u_2).$$

Assume that the following conditions hold:

1. (a₁) $r(x)$ is increasing in $x\in \mathbb {R}^+;$

2. (a₂) $\alpha (x)/\bar {G}(x)$ is increasing in $x\in \mathbb {R}^+;$

3. (a₃) $\tilde {r}(c_1x)/\tilde {r}(c_2x)$ is increasing in $x\in \mathbb {R}^+$ for $0< c_1\le c_2$.

Then, we have

1. (i) $\Xi (x;u_1)\le 0$ for all $x\ge u_1;$

2. (ii) $\Xi (x;u_1)-\Xi (x;u_2)\le 0$ for all $x\ge u_2$.

Proof. (i) The function $\Xi (x;u_1)$ can be rewritten as

$$\Xi(x;u_1)=\alpha(u_1)\alpha(x)F(1;u_1,u_2)F\left(\frac{x}{u_1};u_1,u_2\right) \left(\frac{\bar{F}(\frac{x}{u_1};u_1,u_2)}{F(\frac{x}{u_1};u_1,u_2)\alpha(x)}-\frac{\bar{F}(1;u_1,u_2)}{ F(1;u_1,u_2)\alpha(u_1)}\right),\quad x\ge u_1.$$

On the other hand, we have

\begin{align*} F\left(\frac{x}{u_1};u_1,u_2\right)& =\left[G\left(u_1\frac{x}{u_1}\right)\right]^{\theta_1} \left[G\left(u_2\frac{x}{u_1}\right)\right]^{\theta_2}\\ & =[G(x)]^{\theta_1}\left[G\left(\frac{u_2}{u_1}x\right)\right]^{\theta_2}\\ & =F\left(x;1,\frac{u_2}{u_1}\right),\quad x\in\mathbb{R}^+. \end{align*}

Now, from the above observation and the notion of Lemma 3, one can easily find that

\begin{align*} \Xi(x;u_1)& =\alpha(u_1)\alpha(x)F(1;u_1,u_2)F\left(\frac{x}{u_1};u_1,u_2\right) \left(\frac{\bar{F}(x;1,\frac{u_2}{u_1})}{ F(x;1,\frac{u_2}{u_1})\alpha(x)}-\frac{\bar{F}(u_1;1,\frac{u_2}{u_1})}{ F(u_1;1,\frac{u_2}{u_1})\alpha(u_1)}\right)\\ & =\alpha(u_1)\alpha(x)F(1;u_1,u_2)F\left(\frac{x}{u_1};u_1,u_2\right) \left(L\left(x;\frac{u_2}{u_1}\right)-L\left(u_1;\frac{u_2}{u_1}\right)\right),\quad x\ge u_1. \end{align*}

By Lemma 3, we know $L(x;u_2/u_1)$ is decreasing in $x\in \mathbb {R}^+$ which results in $\Xi (x;u_1)\le 0$ for all $x\ge u_1$.

(ii) Let us define $\mathcal {D}=\{y\geq u_2:\Xi (y;u_2)\le 0\}$. For $x\in \mathcal {D}^{c}$ (the complement of $\mathcal {D}$), we have from Part (i) that $\Xi (x;u_1)-\Xi (x;u_2)\le 0$. In the sequel, it is assume that $x\in \mathcal {D}$. Because $F(x;u_1,u_2)$ is increasing in $x\in \mathbb {R}^+$, it follows that $F(x/u_2;u_1,u_2)\le F(x/u_1;u_1,u_2)$ for all $x\in \mathbb {R}^+$ and $u_2\ge u_1$. Now, using this observation, we have

\begin{align*} & \Xi(x;u_1)-\Xi(x;u_2)\\ & \quad=F\left(\frac{x}{u_1};u_1,u_2\right) F(1;u_1,u_2)\alpha(x)\alpha(u_1)\left(\frac{\bar{F}(\frac{x}{u_1};u_1,u_2)}{ \alpha(x)F(\frac{x}{u_1};u_1,u_2)}-\frac{\bar{F}(1;u_1,u_2)}{\alpha(u_1)F(1;u_1,u_2)}\right) \\ & \qquad -F(\frac{x}{u_2};u_1,u_2) F(1;u_1,u_2)\alpha(x)\alpha(u_2)\left(\frac{\bar{F}(\frac{x}{u_2};u_1,u_2)}{ \alpha(x)F(\frac{x}{u_2};u_1,u_2)}-\frac{\bar{F}(1;u_1,u_2)}{\alpha(u_2)F(1;u_1,u_2)}\right) \\ & \quad\le F\left(\frac{x}{u_1};u_1,u_2\right) F(1;u_1,u_2)\alpha(x)\alpha(u_1)\left(\frac{\bar{F}(\frac{x}{u_1};u_1,u_2)} {\alpha(x)F(\frac{x}{u_1};u_1,u_2)}-\frac{\bar{F}(1;u_1,u_2)}{\alpha(u_1)F(1;u_1,u_2)}\right)\\ & \qquad - F\left(\frac{x}{u_1};u_1,u_2\right) F(1;u_1,u_2)\alpha(x)\alpha(u_2)\left(\frac{\bar{F}(\frac{x}{u_2};u_1,u_2)}{ \alpha(x)F(\frac{x}{u_2};u_1,u_2)}-\frac{\bar{F}(1;u_1,u_2)}{\alpha(u_2)F(1;u_1,u_2)}\right)\\ & \quad= F\left(\frac{x}{u_1};u_1,u_2\right) F(1;u_1,u_2)\alpha(x)\left(\frac{\bar{F}(\frac{x}{u_1};u_1,u_2)}{ \alpha(x)F(\frac{x}{u_1};u_1,u_2)}\alpha(u_1)-\frac{\bar{F}(\frac{x}{u_2};u_1,u_2)}{ \alpha(x)F(\frac{x}{u_2};u_1,u_2)}\alpha(u_2)\right) \\ & \quad=\Upsilon(x),\quad \text{say}. \end{align*}

Setting $c_1=u_1/u_2$ and $c_2=1$ in Condition ($a_3$), it follows that $\tilde {r}(u_1/u_2x)/\tilde {r}(x)\ge \tilde {r}(u_1)/\tilde {r}(u_2)$ or equivalently $\alpha (u_1/u_2x)/\alpha (x)\ge \alpha (u_1)/\alpha (u_2)$ for all $x\geq u_2$. From this observation and Lemma 3, we obtain

\begin{align*} \Upsilon(x)& \stackrel{{\rm sgn}}= \frac{\bar{F}(\frac{x}{u_2};u_1,u_2)}{\alpha(x)F(\frac{x}{u_1};u_1,u_2)}\alpha(u_1) -\frac{\bar{F}(\frac{x}{u_2};u_1,u_2)}{\alpha(\frac{u_1}{u_2}x)F(\frac{x}{u_2};u_1,u_2)} \frac{\alpha(\frac{u_1}{u_2}x)\alpha(u_2)}{\alpha(x)}\\ & \le \frac{\alpha(\frac{u_1}{u_2}x)\alpha(u_2)}{\alpha(x)}\left(\frac{ \bar{F}(\frac{x}{u_1};u_1,u_2)}{\alpha(x)F(\frac{x}{u_1};u_1,u_2)} -\frac{\bar{F}(\frac{x}{u_2};u_1,u_2)}{\alpha(\frac{u_1}{u_2}x)F(\frac{x}{u_2};u_1,u_2)}\right)\\ & =\frac{\alpha(\frac{u_1}{u_2}x)\alpha(u_2)}{\alpha(x)}\left(L\left(x;\frac{u_2}{u_1}\right) -L\left(\frac{u_1}{u_2}x;\frac{u_2}{u_1}\right)\right)\\ & \le 0,\quad x\ge u_2. \end{align*}

Thus, we can conclude that $\Xi (x;u_1)-\Xi (x;u_2)\le 0$ for all $x\ge u_2$, as desired.

For a given point $(\delta _1,\delta _2)$ with $0<\delta _1\le \delta _2$ and $\theta _i\ge 1$, $i=1,2$, we can redefine the region $\Delta _{\theta _1,\theta _2}(\delta _1,\delta _2)$, proposed in Introduction, as follows:

$$\Delta_{\theta_1,\theta_2}(\delta_1,\delta_2)=\{(x,y)\in \mathbb{R}^{{+}2}:\, \delta_1 \le x\le y\le \delta_2\ \text{and}\ \theta_1\delta^{{-}1}_1+\theta_2\delta^{{-}1}_2\ge\theta_1 x^{{-}1}+\theta_2 y^{{-}1}\}.$$

A question arises here: what condition the function $\varphi :{\mathbb {R}^+}^2\rightarrow \mathbb {R}$ must have to satisfy the inequality $\varphi (\delta _1,\delta _2)\ge \varphi (\delta ^{*}_1,\delta ^{*}_2)$ for $(\delta ^{*}_1,\delta ^{*}_2)\in \Delta _{\theta _1,\theta _2}(\delta _1,\delta _2)?$ If $\theta _1{\delta ^{*}}^{-1}_1+\theta _2 {\delta ^{*}}^{-1}_2=\theta _1\delta ^{-1}_1+\theta _2\delta ^{-1}_2$, then the point $(\delta ^{*}_1,\delta ^{*}_2)$ lies on the curve $\theta _1 x^{-1}+\theta _2 y^{-1}=\theta _1\delta ^{-1}_1+\theta _2\delta ^{-1}_2$. Because the vector field of this curve is $(1,-\theta _1\theta ^{-1}_2x^{-2}y^{2})$, then the inequality $\varphi (\delta _1,\delta _2)\ge \varphi (\delta ^{*}_1,\delta ^{*}_2)$ holds if $\varphi (x,y)$ is decreasing along the vector $(1,-\theta _1\theta ^{-1}_2x^{-2}y^{2})$. If $(\delta ^{*}_1,\delta ^{*}_2)$ lies inside the region $\Delta _{\theta _1,\theta _2}(\delta _1,\delta _2)$, that is, $\theta _1{\delta ^{*}}^{-1}_1+\theta _2 {\delta ^{*}}^{-1}_2<\theta _1\delta ^{-1}_1+\theta _2\delta ^{-1}_2$, then there exists a point $(\delta '_1,\delta ^{*}_2)$ on the curve $\theta _1 x^{-1}+\theta _2 y^{-1}=\theta _1\delta ^{-1}_1+\theta _2\delta ^{-1}_2$ such that $\delta _1<\delta '_1$. In this case, if $\varphi (x,y)$ is decreasing along the vectors $(1,0)$ and $(1,-\theta _1\theta ^{-1}_2x^{-2}y^{2})$, then we have $\varphi (\delta _1,\delta _2)\ge \varphi (\delta '_1,\delta ^{*}_2)\ge \varphi (\delta ^{*}_1,\delta ^{*}_2)$. Consequently, we can state the following lemma.

Lemma 5. Consider the function $\varphi :{\mathbb {R}^+}^2\rightarrow \mathbb {R}$. If $\varphi (u_1,u_2)$ is decreasing along the vectors $\boldsymbol {v}_1=(1,0)$ and $\boldsymbol {v}_2=(1,-\theta _1\theta ^{-1}_2u^{-2}_1 u^{2}_2)$, then we have

$$(\delta^{*}_1,\delta^{*}_2)\in\Delta_{\theta_1,\theta_2}(\delta_1,\delta_2) \Rightarrow\varphi(\delta_1,\delta_2)\ge\varphi(\delta^{*}_1,\delta^{*}_2).$$

Next theorem deals with the mean residual life order between the largest order statistics arising from the RS models.

Theorem 1. Suppose $(T_1,\ldots,T_n)\sim RS(G;\boldsymbol {\theta },\boldsymbol {\delta }_{l})$ and $(T^{*}_1,\ldots,T^{*}_n)\sim RS(G;\boldsymbol {\theta },\boldsymbol {\delta }^{*}_{l})$. Set $\xi _1=\sum _{i=1}^{l}\theta _i$ and $\xi _{2}=\sum _{i=l+1}^{n}\theta _i$. Assume that the following conditions hold:

1. (a₁) $r(x)$ is increasing in $x\in \mathbb {R}^+$;

2. (a₂) $\alpha (x)/\bar {G}(x)$ is increasing in $x\in \mathbb {R}^+$;

3. (a₃) $\tilde {r}(c_1x)/\tilde {r}(c_2x)$ is increasing in $x\in \mathbb {R}^+$ for $0< c_1\le c_2$.

If $(\delta ^{*}_1,\delta ^{*}_2)\in \Delta _{\xi _1,\xi _2}(\delta _1,\delta _2)$, then we have $T_{n:n}\ge _{{\rm mrl}}T^{*}_{n:n}$.

Proof. The distribution functions of $T_{n:n}$ and $T^{*}_{n:n}$ are respectively as

$$F_{T_{n:n}}(x)=[G(\delta_1x)]^{\xi_1}[G(\delta_2x)]^{\xi_2},\quad F_{T^{*}_{n:n}}(x)=[G(\delta^{*}_1x)]^{\xi_1}[G(\delta^{*}_2x)]^{\xi_2},\quad x\in \mathbb{R}^+.$$

Evidently, the distribution function of $T_{n:n}$ (resp. $T^{*}_{n:n}$) is the same as that of $T_{2:2}$ (resp. $T^{*}_{2:2}$) by replacing $\theta _i$ by $\xi _i$ for $i=1,2$. Therefore, it is enough to prove the required result for the special case of $n=2$. The mean residual life functions of $T_{2:2}$ and $T^{*}_{2:2}$ can be rewritten respectively as

$$m_{T_{2:2}}(x)= x\varphi(\delta_1 x,\delta_2 x), \quad m_{T^{*}_{2:2}}(x)=x\varphi(\delta^{*}_1 x,\delta^{*}_2 x),\quad x\in \mathbb{R}^+,$$

wherein

$$\varphi(u_1,u_2)=\frac{\int_1^{\infty}\bar{F}(x;u_1,u_2)\,dx}{\bar{F}(1;u_1,u_2)},\quad 0< u_1\leq u_2.$$

According to Lemma 5, the desired result follows if we could show that $\varphi (u_1,u_2)$ is decreasing at the directions $\pmb {v}_1$ and $\pmb {v}_2$. The partial derivative of $\varphi (u_1,u_2)$ with respect to $u_1$ can be expressed as

\begin{align*} \partial_1\varphi(u_1,u_2)& =[\bar{F}(1;u_1,u_2)]^{{-}2} \left\{-\theta_1\bar{F}(1;u_1,u_2)\int_1^{\infty} x\tilde{r}(u_1 x)F(x;u_1,u_2)\,dx\right.\\ & \quad \left.+\theta_1 \tilde{r}(u_1)F(1;u_1,u_2) \int_1^{\infty}\bar{F}(x;u_1,u_2)\,dx\right\} \\ & =\theta_1u_1^{{-}2}[\bar{F}(1;u_1,u_2)]^{{-}2}\int_{u_1}^{\infty}\Xi(x;u_1)\,dx, \end{align*}

wherein the function $\Xi (.;u_1)$ is defined in Lemma 4. Similarly, the partial derivative of $\varphi (u_1,u_2)$ with respect to $u_2$ is

$$\partial_2 \varphi(u_1,u_2)=\theta_2u_2^{{-}2} [\bar{F}(1;u_1,u_2)]^{{-}2}\int_{u_2}^{\infty}\Xi(x;u_2)\,dx.$$

Using Part (i) of Lemma 4, one can easily observe that $\Xi (x;u_1) \le 0$ for all $x\ge u_1$. Hence, we find that

$$\nabla_{\pmb{v}_1}\varphi=\partial_1\varphi(u_1,u_2)\leq 0,$$

and so, $\varphi$ is decreasing at the direction $\pmb {v}_1$. The gradient of $\varphi$ along the vector $\pmb {v}_2$ is

\begin{align*} \nabla_{\pmb{v_2}}\varphi& =\partial_1\varphi(u_1,u_2)-\theta_1\theta^{{-}1}_2u^{2}_2u^{{-}2}_1\partial_2\varphi(u_1,u_2)\\ & \stackrel{{\rm sgn}}=\int_{u_1}^{\infty}\Xi (x;u_1)\,dx-\int_{u_2}^{\infty}\Xi(x;u_2)\,dx\\ & =\int_{u_1}^{u_2}\Xi(x;u_1)\,dx+\int_{u_2}^{\infty}(\Xi(x;u_1)-\Xi(x;u_2))\,dx \end{align*}

According to Lemma 4, it readily follows $\nabla _{\boldsymbol {v}_2}\varphi \le 0$ which results in $\varphi$ is also decreasing at the direction $\boldsymbol {v}_2$, completing the proof of the theorem.

In Theorem 1, the inference is focused on the scale parameters while both involved vectors of random variables have the common resilience parameters. It is worthwhile to point that the effect of resilience parameters on the mean residual life function of $T_{n:n}$ is also an interesting problem. To see more information in this direction, we refer the readers to Haidari et al. [17]. It should be mentioned that the result of Theorem 1 extends those of Zhao and Balakrishnan [33] and Wang and Cheng [29] which are established when the baseline distribution is exponential.

Now, let us give a reliability explanation of Theorem 1. Consider a factory that produces some specific units with parallel structures made up $n$ components. Suppose the components used in building the units come from a supplier, say Supplier I, which has two production lines. Due to production policies, the factory selects $l$ components from one of the production line and the remaining $n-l$ components from the another one. Supplier I asserts the lifetimes of its produced components in each line follow the RS models with same scale parameters but with possibly different resilience parameters. For some reasons such as high price or unavailability of the components in a specific period of time, the factory decides to purchase its required components from a new supplier, say Supplier II. The produced components by Supplier II, like Supplier I, are built in two production lines with their lifetimes following the RS models with same scale parameters but with possibly different resilience parameters. In such a case, changing the components may impress the quality of the units of the factory. Therefore, to avoid the quality loss of the units, the factory must investigate the effect of these changes. In this situation, Theorem 1 gives some sufficient conditions to compare the mean residual life functions of the units comprising the components of Suppliers I and II.

Next proposition is an immediate consequence of Theorem 1 because the scale model can be obtained from the RS model when all the resilience parameters are equal to 1.

Proposition 1. Consider two sets of independent nonnegative random variables $T_1,\ldots,T_n$ and $T^{*}_1,\ldots,T^{*}_n$ following the multiple-outlier scale models with common baseline distribution function $G$ and respective vectors of scale parameters $(\delta _1\boldsymbol {1}_{l},\delta _2\boldsymbol {1}_{n-l})$ and $(\delta ^{*}_1\boldsymbol {1}_{l},\delta ^{*}_2\boldsymbol {1}_{n-l})$. Assume that the following conditions hold:

1. (a₁) $r(x)$ is increasing in $x\in \mathbb {R}^+$;

2. (a₂) $\alpha (x)/\bar {G}(x)$ is increasing in $x\in \mathbb {R}^+$;

3. (a₃) $\tilde {r}(c_1x)/\tilde {r}(c_2x)$ is increasing in $x\in \mathbb {R}^+$ for $0< c_1\le c_2$.

If $(\delta ^{*}_1,\delta ^{*}_2)\in \Delta _{l,n-l}(\delta _1,\delta _2)$, then we have $T_{n:n}\ge _{{\rm mrl}}T^{*}_{n:n}$.

Remark 1. It should be noted that Condition ($a_2$) in Theorem 1 (Proposition 1) satisfies if and only if

$$1+\gamma(x)+xr(x)\ge 0,\quad \text{for all}\ x\in\mathbb{R}^+,$$

which, by Eqs. (1) and (2), can be rewritten as

$$1+\frac{x r'(x)}{r(x)}-\alpha(x)\ge 0,\quad\text{for all}\ x\in\mathbb{R}^+.$$

Furthermore, Condition ($a_3$) in Theorem 1 (Proposition 1) is equivalent to say that $\gamma (x)$ is decreasing in $x\in \mathbb {R}^+$.

3. Illustration with examples

In this section, we present some examples of well-known distributions verifying the conditions of the results given in the previous section. Recall that $\bar {G}$, $r$ and $\tilde {r}$ are respectively the survival, hazard rate and reversed hazard rate functions of the baseline distribution in both RS and scale models. Also, the functions $\alpha$ and $\gamma$ are defined as $\alpha (x)=x\tilde {r}(x)$ and $\gamma (x)=x\tilde {r}'(x)/\tilde {r}(x)$ for $x\in \mathbb {R}^+$.

3.1. Exponentiated generalized gamma distribution

If a random variable $Y$ admits the following distribution function

$$F(x;\tau,\beta,\theta,\delta)=\left[\int_{0}^{x}\frac{\tau\delta^{\beta}}{\Gamma(\frac{\beta}{\tau})}u^{\beta-1} e^{-(\delta u)^{\tau}}\,du\right]^{\theta},\quad x\in \mathbb{R}^+,\ (\theta,\tau,\beta,\delta)\in \mathbb{R}^{{+}4},$$

wherein $\Gamma (\cdot )$ is the incomplete gamma function, then it is said that $Y$ has the exponentiated generalized gamma (EGG) distribution with shape parameters $(\theta,\tau,\beta )$ and scale parameter $\delta$, denoted by $Y\sim {\rm EGG}(\theta,\tau,\beta,\delta )$. This distribution is introduced and investigated comprehensively by Cordeiro et al. [10]. The EGG distribution contains some known distributions such as Weibull, generalized gamma (GG), generalized exponential, exponentiated Weibull and exponentiated gamma as special cases. Note that, the EGG distribution belongs to the RS model when the baseline distribution is the GG distribution with shape parameters $(\tau,\beta )$ and scale parameter 1 (denoted by ${\rm GG}(\tau,\beta )$ and called as the GG distribution with shape parameters $(\tau,\beta )$); see Kleiber and Kotz [19] for more details on the GG distribution and its applications.

To show Theorem 1 can be applied for the EGG distribution, we need the following lemma.

Lemma 6. The GG distribution with shape parameters $(\tau,\beta )$ satisfies all conditions of Theorem 1 when $\tau \ge \beta \ge 1$.

Proof. Khaledi et al. [18] proved the followings for ${\rm GG}(\tau,\beta )$:

(5) \begin{align} \alpha(x)& \le\beta,\quad x\in\mathbb{R}^+; \end{align}

(6) \begin{align} \beta-1& < x\frac{r'(x)}{r(x)}<\tau-1,\quad x\in\mathbb{R}^+,\ \tau>\beta. \end{align}

When $\tau \ge \beta \ge 1$, it is well-known that the ${\rm GG}$ distribution has an increasing hazard rate function, and so, Condition ($a_1$) of Theorem 1 is fulfilled. Also, upon combining Eqs. (5) and (6), we find that $1+(x r'(x))/r(x)-\alpha (x)\ge 0$ for all $x\in \mathbb {R}^+$ and $\tau >\beta$. Using this observation and Remark 1, we see that $\alpha (x)/\bar {G}(x)$ is increasing in $x\in \mathbb {R}^+$ when $\tau >\beta$. Furthermore, if $\tau =\beta$, then the ${\rm GG}$ distribution is reduced to Weibull distribution. In this case, we have $\alpha (x)/\bar {G}(x)=\beta x^{\beta }e^{x^{\beta }}$ which clearly is increasing in $x\in \mathbb {R}^+$ for all $\beta \in \mathbb {R}^+$. Hence, Condition ($a_2$) of Theorem 1 is satisfied for $\tau \ge \beta$. As Ding et al. [12] have shown, $\gamma (x)$ is decreasing in $x\in \mathbb {R}^+$ for all $(\tau,\beta )\in {\mathbb {R}^+}^2$ and so, by Remark 1, we can conclude that Condition ($a_3$) of Theorem 1 is held. The proof is now completed.

Next corollary is a direct consequence of Theorem 1 and Lemma 6.

Corollary 1. Let $T_1,\ldots,T_n$ and $T^{*}_1,\ldots,T^{*}_n$ be two sets of independent random variables with $T_i\sim {\rm EGG}(\theta _i,\tau,\beta,\delta _1)$, $T_j\sim {\rm EGG}(\theta _j,\tau,\beta,\delta _2)$, $T^{*}_i\sim {\rm EGG}(\theta _i,\tau,\beta,\delta ^{*}_1)$ and $T^{*}_j\sim {\rm EGG}(\theta _j,\tau,\beta,\delta ^{*}_2)$ for $i=1,\ldots,l$ and $j=l+1,\ldots,n$. If $\tau \ge \beta \ge 1$ and $(\delta ^{*}_1,\delta ^{*}_2)\in \Delta _{\xi _1,\xi _2}(\delta _1,\delta _2)$, then we have $T_{n:n}\ge _{{\rm mrl}}T{^*}_{n:n}$.

Several well-known lifetime distributions satisfy in Corollary 1 as listed below:

1. (i) Set $\theta _1=\cdots =\theta _n=1$ and $\tau =\beta =1$. In this case, we have the multiple-outlier exponential models. This statement is proved by Zhao and Balakrishnan [33] for $n=2$, while the general case is established by Wang and Cheng [29].

2. (ii) Set $\theta _1=\cdots =\theta _n=1$ and $\tau =\beta$. This case deals with Weibull distributed random variables with common shape parameter $\tau$ and reduced scale parameters. Note that, the mean residual life order holds under the restriction $\tau \ge 1$. For the special case of $\tau =2$, Rayleigh distribution is involved.

3. (iii) Set $\tau =\beta =1$. In this statement, we have the generalized exponential distributed random variables with heterogeneous shape parameters and reduced scale parameters.

4. (iv) Set $\tau =\beta$. This case concerns the exponentiated Weibull distributed random variables. Like Case (ii), a restriction $\tau \ge 1$ is appeared for the mean residual life order to be hold.

3.2. Power-generalized Weibull distribution

A random variable $Y$ is said to have the power-generalized Weibull (PGW) distribution with shape parameters $(\rho,\gamma )$ and scale parameter $\delta$, denoted by ${\rm PGW}(\rho,\gamma,\delta )$, if its distribution function is as follows:

$$F(x;\rho,\gamma,\delta)=1-e^{1-(1+(\delta x)^{\rho})^{{1}/{\gamma}}},\quad x\in\mathbb{R}^+, (\rho,\gamma,\delta)\in{\mathbb{R}^+}^3.$$

This distribution is introduced by Bagdonavicius and Nikulin [2] in the context of accelerated failure time models. It contains exponential, Rayleigh and Weibull distributions as special cases. The PGW distribution is a suitable model to analysis the lifetime data sets due to its flexible hazard rate function which admits monotone and non-monotone shapes; see Nikulin and Haghighi [26] and Nadaraja and Haghighi [25]. It is clear that the PGW distribution belongs to the scale model with the baseline distribution as ${\rm PGW}(\rho,\gamma,1)$ (called as the PGW distribution with shape parameters $(\rho,\gamma )$).

Before presenting an application of Proposition 1 for the case of PGW distribution, we state the next lemma.

Lemma 7. The PGW distribution with shape parameters $(\rho,\gamma )$ satisfies all conditions of Proposition 1 when $\rho \ge 1$ and $\gamma \le 1$.

Proof. With the restrictions $\rho \ge 1$ and $\gamma \le 1$, Condition ($a_1$) of Proposition 1 is satisfied because the PGW distribution admits an increasing hazard rate function under the mentioned restrictions. Furthermore, as shown by Ding et al. [12], we have

$$x\frac{r'(x)}{r(x)}=\rho-1+\rho\left(\frac{1}{\gamma}-1\right)\frac{x^{\rho}}{1+x^{\rho}},\quad x\in\mathbb{R}^+.$$

It is obvious that $(x r'(x))/r(x)$ is increasing in $x\in \mathbb {R}^+$ for $\gamma \le 1$. Also, one can easily find that

$$\lim_{x\rightarrow 0} x\frac{r'(x)}{ r(x)}=\rho-1,\quad \lim_{x\rightarrow\infty} x\frac{r'(x)}{ r(x)}=\frac{\rho}{\gamma}-1.$$

Upon combining the above observations, we obtain

(7) \begin{equation} \rho-1\le x\frac{r'(x)}{r(x)}\le\frac{\rho}{\gamma}-1,\quad x\in\mathbb{R}^+,\ \gamma\le 1. \end{equation}

Furthermore, using the L'Hopital's rule, it follows that

(8) \begin{align} \lim_{x\rightarrow 0}\alpha(x)& =\frac{\rho}{\gamma}\lim_{x\rightarrow 0} \frac{x^{\rho}(1+x^{\rho})^{{1}/{\gamma}-1}}{e^{(1+x^{\rho})^{{1}/{\gamma}}-1}-1}\nonumber\\ & =\lim_{x\rightarrow 0}\frac{\rho (1+x^{\rho})+\rho(\frac{1}{\gamma}-1)x^{\rho}} {(1+x^{\rho})e^{(1+x^{\rho})^{{1}/{\gamma}}-1}}\nonumber\\ & =\rho. \end{align}

According to Lemma A.6 of Khaledi et al. [18], we know that $\alpha (x)$ is decreasing in $x\in \mathbb {R}^+$ which along with Eq. (8) result in

(9) \begin{equation} \alpha(x)\le\rho,\quad x\in\mathbb{R}^+. \end{equation}

Now, using Eqs. (7) and (9), we readily find that $1+(x r'(x))/r(x)-\alpha (x)\ge 0$ for all $x\in \mathbb {R}^+$ and $\gamma \le 1$ and so, based on Remark 1, one can observe that $\alpha (x)/\bar {G}(x)$ is increasing in $x\in \mathbb {R}^+$ when $\gamma \le 1.$ Hence, Condition ($a_2$) of Proposition 1 is fulfilled under the restriction $\gamma \le 1$. Furthermore, from Remark 1 once again and Lemma 4.11 of Ding et al. [12], we see that Condition ($a_3$) of Proposition 1 is satisfied when $\gamma \le 1$, completing the proof of the lemma.

From Proposition 1 and Lemma 7, the next corollary readily follows.

Corollary 2. Let $T_1,\ldots,T_n$ and $T^{*}_1,\ldots,T^{*}_n$ be two sets of independent random variables with $T_i\sim {\rm PGW}(\rho,\gamma,\delta _1)$, $T_j\sim {\rm PGW}(\rho,\gamma,\delta _2)$, $T^{*}_i\sim {\rm PGW}(\rho,\gamma,\delta ^{*}_1)$ and $T^{*}_j\sim {\rm PGW}(\rho,\gamma,\delta ^{*}_2)$ for $i=1,\ldots,l$ and $j=l+1,\ldots,n$. If $\rho \ge 1$, $\gamma \le 1$ and $(\delta ^{*}_1,\delta ^{*}_2)\in \Delta _{l,n-l}(\delta _1,\delta _2)$, then we have $T_{n:n}\ge _{{\rm mrl}}T^{*}_{n:n}$.

Next example illustrates the result given in Corollary 2.

Example 1. Set $n=4$, $l=2$, $\rho =1$, $\gamma =0.5$, $(\delta _1,\delta _2)=(2,3.5)$ and $(\delta ^{*}_1,\delta ^{*}_2)=(3,3.2)$. In this case, we have

$$\Delta_{2,2}(2,3.5)=\left\{(x,y)\in{\mathbb{R}^+}^2:\,2\le x\le y\le 3.5\ \text{and}\ x^{{-}1}+y^{{-}1}\le\frac{11}{14}\right\}.$$

It can be readily seen that $(\delta ^{*}_1,\delta ^{*}_2)\in \Delta _{2,2}(2,3.5)$ which, according to Corollary 2, results in $T_{4:4}\ge _{{\rm mrl}}T^{*}_{4:4}$. In Figure 2, the mean residual life functions of $T_{4:4}$ and $T^{*}_{4:4}$ is plotted over the interval $(0,1.1]$.

Figure 2. Plot of the mean residual functions of $T_{4:4}$ and $T^{*}_{4:4}$ when $l=2$, $\rho =1$, $\gamma =0.5$, $(\delta _1,\delta _2)=(2,3.5)$ and $(\delta ^{*}_1,\delta ^{*}_2)=(3,3.2)$ for random variables with PGW distributions.

3.3. Half-normal distribution

Consider a random variable $Y$ with its distribution function has the following form:

$$F(x;\lambda)=\int_{0}^{x}\frac{\delta\sqrt{2}}{\sqrt{\pi}} e^{-{(\delta y)^2}/{2}}\,dy,\quad x\in\mathbb{R}^+,\delta\in\mathbb{R}^+.$$

Then, it is said that $Y$ follows the half-normal (HN) distribution with scale parameter $\delta$, denoted by $Y\sim HN(\delta )$. Clearly, the HN distribution belongs to the scale model and the baseline distribution in this case obtains by choosing $\delta =1$ (called as the standard HN).

Lemma 8. The standard HN satisfies all conditions of Proposition 1.

Proof. The hazard function of the standard HN is as follows:

$$r(x)=\frac{e^{-{x^2}/{2}}}{\int_{x}^{\infty}e^{-{y^2}/{2}}\,dy},\quad x\in\mathbb{R}^+.$$

Taking derivative of $r(x)$ with respect to $x\in \mathbb {R}^+$, we find that

\begin{align*} r'(x)& \stackrel{{\rm sgn}}=e^{-{x^2}/{2}}-x\int_{x}^{\infty}e^{-{y^2}/{2}}\,dy\\ & =s(x),\quad \text{say}. \end{align*}

It is easy to see that $s'(x)\le 0$ for all $x\in \mathbb {R}^+$. From this observation and the fact that $\lim _{x\rightarrow \infty }s(x)=0$, we can readily conclude that $r(x)$ is increasing in $x\in \mathbb {R}^+$. Thus, Condition ($a_1$) of Proposition 1 is held. Moreover, from Ding et al. [12], we have

$$\gamma(x)={-}x^{2}-\alpha(x),\quad x\in\mathbb{R}^+,$$

which, for all $x\in \mathbb {R}^+$, results in

\begin{align*} 1+\gamma(x)+xr(x)& =(1-\alpha(x))+x(r(x)-x)\\ & =D_1+D_2, \quad \text{say}. \end{align*}

As pointed out by Ding et al. [12], we know $\alpha (x)\le 1$ for all $x\in \mathbb {R}^+$ and hence $D_1\ge 0$. Furthermore, it is shown in the above that $s(x)\ge 0$ or equivalently $r(x)\ge x$ for all $x\in \mathbb {R}^+$ which results in $D_2\ge 0$. Therefore, $1+\gamma (x)+xr(x)\ge 0$ for all $x\in \mathbb {R}^+$ and so, according to Remark 1, Condition ($a_2$) of Proposition 1 is also satisfied. Finally, from Ding et al. [12] and Remark 1, we can see that Condition ($a_3$) of Proposition 1 is obtained. The proof is now completed.

From Proposition 1 and Lemma 8, the next corollary immediately follows.

Corollary 3. Let $T_1,\ldots,T_n$ and $T^{*}_1,\ldots,T^{*}_n$ be two sets of independent random variables with $T_i\sim HN(\delta _1)$, $T_j\sim HN(\delta _2)$, $T^{*}_i\sim HN(\delta ^{*}_1)$ and $T^{*}_j\sim HN(\delta ^{*}_2)$ for $i=1,\ldots,l$ and $j=l+1,\ldots,n$. If $(\delta ^{*}_1,\delta ^{*}_2)\in \Delta _{l,n-l}(\delta _1,\delta _2)$, then we have $T_{n:n}\ge _{{\rm mrl}}T^{*}_{n:n}$.

In the following example, the result given in Corollary 3 is investigated numerically.

Example 2. Set $n=2$, $l=1$, and $(\delta _1,\delta _2)=(0.5,1)$, $(\delta ^{*}_1,\delta ^{*}_2)=(0.75,0.8)$. Then, it is easy to see that $(\delta ^{*}_1,\delta ^{*}_2)\in \Delta _{1,1}(0.5,1)$ wherein

$$\Delta_{1,1}(0.5,1)=\{(x,y)\in{\mathbb{R}^+}^2:\,0.5\le x\le y\le 1\ \text{and}\ x^{{-}1}+y^{{-}1}\le 3\}.$$

Now, by Corollary 3, we can conclude $T_{2:2}\ge _{{\rm mrl}}T^{*}_{2:2}$. In Figure 3, the mean residual life functions of $T_{2:2}$ and $T^{*}_{2:2}$ is plotted over the interval $(0,9]$.

Figure 3. Plot of the mean residual functions of $T_{2:2}$ and $T^{*}_{2:2}$ when $l=1$, $(\delta _1,\delta _2)=(0.5,1)$ and $(\delta ^{*}_1,\delta ^{*}_2)=(0.75,0.8)$ for random variables with HN distributions.

4. Discussion

In this paper, we have established some conditions for the mean residual life order between the largest order statistics in the RS models to be hold. We have also studied an application of this result for the case of exponentiated generalized gamma, power-generalized Weibull and half-normal distributions. The results established here extend and reinforce those of Zhao and Balakrishnan [33] and Wang and Cheng [29]. Following the definitions of weighted majorization and related orders presented in Cheng [9], let us define the weighted version of reciprocal majorization order. Set $\mathcal {E}^+_{n}=\{(x_1,\ldots,x_n)\in {\mathbb {R}^+}^n:0< x_1\le \dots \le x_n\}$ and $\boldsymbol {\theta }=(\theta _1,\ldots,\theta _n)\in \mathbb {R}^n$. For two vectors $(u_1,\ldots,u_n)$ and $(v_1,\ldots,v_n)$ in $\mathcal {E}^+_{n}$, we say that $(u_1,\ldots,u_n)$ is greater than $(v_1,\ldots,v_n)$ on $\mathcal {E}^+_n$ with respect to $\boldsymbol {\theta }$-reciprocal majorization order, denoted by $(u_1,\ldots,u_n)\stackrel {{\rm rm}}{\succ }_{\boldsymbol {\theta }}(v_1,\ldots,v_n)$ on $\mathcal {E}^+_n$, if $\sum _{k=1}^{i}\theta _ku^{-1}_k\ge \sum _{k=1}^{i}\theta _kv^{-1}_k$ for all $i=1,\ldots,n$. In the definition of the region $\Delta _{\theta _1,\theta _2}(\delta _1,\delta _2)$, the restrictions $\theta _1\ge 1$ and $\theta _2\ge 1$ are appeared because of utilizing the inequality in Lemma 1. Now, if $(\delta ^{*}_1,\delta ^{*}_2)\in \Delta _{\theta _1,\theta _2}(\delta _1,\delta _2)$ without any restriction on $\theta _i$'s, then we can see that $\delta _1\le \delta ^{*}_1\le \delta ^{*}_2\le \delta _2$ and $(\delta _1,\delta _2)\stackrel {{\rm rm}}{\succ }_{(\theta _1,\theta _2)}(\delta ^{*}_1,\delta ^{*}_2)$ on $\mathcal {E}^+_2$. Thus, if the restrictions are dropped, then the weighted version of reciprocal majorization order can be used in comparison of the largest order statistics in the RS models. With the help of some numerical examples, we conjecture that the mean residual life order given in Theorem 1 may hold without any restrictions on $\theta _i$'s. We are currently working on this problem and hope to report the findings in the future works.