1. Introduction and preliminaries
Most often, the real-world populations of items are heterogeneous and the corresponding homogeneity can be considered as some approximation. There can be different reasons for heterogeneity. For example, the items can be produced by different manufacturers and then mixed by the user. It can happen with one manufacturer as well, as reliability characteristics of manufactured items change with time depending on many factors (e.g. supplied material, human factors, the production condition, etc.). Heterogeneity can be also induced by a random environment in which items are operating. This random environment can be modelled by the corresponding shock process, for example (see e.g. [Reference Cha and Finkelstein8] and the references therein). However, the simplest and sometimes the most effective way is to model it by the non-negative environmental random variable
$\Theta$
that can affect the distribution of a lifetime X in a baseline deterministic environment (denoted by
$F_X(x)$
). Thus, conditional on a realization
$\Theta=\theta$
, the corresponding distribution of
$X(\theta)$
is
$F_X(x\!{\mid}\!\theta)$
, where the most popular specific models are the scale model
$F_X(\theta x)$
, the multiplicative and additive frailty models written as
$\theta r_X(x)$
and
$r_X(x)+\theta$
, where
$r_X(x)$
is the corresponding failure rate assuming that it exists.
The most common example of environment is a stress or load under which technical systems are operating (e.g. an electrical load). Some overall climate or nutrition parameters can also describe environment for organisms. An effect of a random environment on various reliability indices was intensively studied in the literature: see e.g. [Reference Finkelstein11], [Reference Kenzin and Frostig16], [Reference Nakagawa22], [Reference Persona, Sgarbossa and Pham30], [Reference Petakos and Tsapelas31], [Reference Rde34], and the references therein.
While describing the following baseline simplified scenarios to be considered in our paper in much more generality, we will use several basic stochastic orders, to be defined for convenience by Definition 1.1 at the end of this section.
(i) Consider two items (systems) with lifetimes
$X(\Theta)$
and
$Y(\Theta)$
(later we will use a slightly different notation that is more appropriate for the multicomponent case) operating in the same environment modelled by
$\Theta$
, and we are interested in stochastic comparisons of these lifetimes. Note that they are dependent via the common environment. Obviously, if we know that its impact is the same on both items (e.g. the multiplicative frailty model), then in order for
$X(\Theta ){{\le }_{\text{st}}}Y(\Theta )$
, for example, to hold, it is sufficient for this inequality to be true for the baseline, deterministic environment, i.e.
$X\leq_{\text{st}} Y$
, as it will hold in each realization of
$\Theta$
. For a general case, we must just assume this property in each realization, i.e.
$X(\theta)\leq_{\text{st}}Y(\theta)$
, for all
$\theta.$
(ii) Now let one item (or two statistically identical items) operate in two environments
$\Theta_1$
and
$\Theta_2$
with the corresponding lifetimes
$X(\Theta_1)$
and
$X(\Theta_2)$
. There are a number of simple, meaningful results in the literature for the corresponding comparisons. For example, in [Reference Finkelstein12] and [Reference Shaked and Shanthikumar37] it is stated that if
$\Theta_1\leq_{\text{hr}}\Theta_2$
, then
$X(\Theta_1)\leq_{\text{hr}}X(\Theta_2)$
, provided that the corresponding failure rate
$r(t\!{\mid}\!\theta)$
is ordered in
$\theta$
for all
$t>0$
.(iii) The last general introductory setting to be considered is when two items with lifetimes
$X(\Theta_1)$
and
$Y(\Theta_2)$
, are operating in different environments modelled by
$\Theta_1$
and
$\Theta_2$
, respectively. A specific case is when, for example,
$\Theta_2$
is degenerate, meaning that the second environment is deterministic.
The above scenarios are described with respect to comparisons of lifetimes of two items or systems with a black box description. However, our paper deals with these scenarios for the multicomponent systems, namely, coherent systems that satisfy two basic requirements: each component is important for operation of a system and the system lifetime should not decrease if we replace any failed component with a ‘new’ one. This class of systems is rather wide and includes, for example, the k-out-of-n systems as a special case (see [Reference Barlow and Proschan5]).
Stochastic comparisons for k-out-of-n systems with independent components are extensively studied in [Reference Balakrishnan and Zhao4], [Reference Hazra, Kuiti, Finkelstein and Nanda13], [Reference Pledger, Proschan and Rustagi32], and [Reference Proschan and Sethuraman33], to name a few. General coherent systems with independent components are considered, for example, in [Reference Belzunce, Franco, Ruiz and Ruiz6], [Reference Esary and Proschan10], [Reference Hazra and Nanda14], [Reference Kochar, Mukerjee and Samaniego17], [Reference Lindqvist, Samaniego and Huseby18], [Reference Nanda, Jain and Singh23], [Reference Samaniego35], and [Reference Samaniego and Navarro36]. Stochastic ordering for coherent systems with dependent components is discussed in [Reference Navarro, Águila, Sordo and Suárez-Liorens24], [Reference Navarro, Águila, Sordo and Suárez-Liorens25], [Reference Navarro, Águila, Sordo and Suárez-Liorens26], [Reference Navarro, Pellerey and Di Crescenzo27], and [Reference Navarro and Rubio28]. The impact of a random environment has been studied via multivariate mixture models; references include [Reference Bada, Sangüesa and Cha2], [Reference Balakrishnan, Barmalzan and Haidari3], [Reference Belzunce, Mercader, Ruiz and Spizzichino7], [Reference Marshall and Olkin19], and [Reference Misra and Misra20].
However, to the best of our knowledge, only two papers have been devoted to stochastic comparisons of coherent systems with dependent components operating in different random environments [Reference Amini-Seresht, Zhang and Balakrishnan1, Reference Mulero, Pellery and Rodríguez-Griñolo21]. These publications provide some detailed sufficient conditions for the lifetime of a coherent system operating in one environment (e.g. the more severe) to be smaller than that of this system operating in a milder environment. This setting generalizes scenario (ii) above to the case of systems with dependent components. It should be noted that there are two ‘combined’ sources of this dependence. One results from the dependence of the components in the coherent system operating in a deterministic environment and the other is due to the dependence of components of the system sharing the same random environment.
Inspired by the work of these authors, we present solutions for some open problems formulated in [Reference Amini-Seresht, Zhang and Balakrishnan1]. We also generalize their results and present some new comparisons which consider the settings (i) and (iii) applied to coherent systems with dependent components. To be more specific, we provide different sets of sufficient conditions for one system to dominate the other with respect to different stochastic orders, namely, usual stochastic order, hazard rate order, reversed hazard rate order, and likelihood ratio order. It is worth mentioning that the above scenario (ii) considered in [Reference Amini-Seresht, Zhang and Balakrishnan1] for coherent systems can be viewed as a specific case of a more general scenario (iii). Moreover, our methodology for obtaining relevant comparisons also differs from that discussed in [Reference Amini-Seresht, Zhang and Balakrishnan1]. Lastly, although the case when
$\Theta_2$
is degenerate (meaning that the second environment is deterministic) is specific for scenario (iii), for technical reasons it is convenient to consider it separately, which is done in Section 5.
After this introductory discussion, we can carry on with some basic facts to be used intensively throughout the paper, starting with the relevant, formal notation that is more convenient for our multivariate setting. For an absolutely continuous random variable W, we denote the probability density function (pdf) by
$f_W(\!\cdot\!)$
, the cumulative distribution function (cdf) by
$F_W(\!\cdot\!)$
, the hazard rate function by
$r_W(\!\cdot\!)$
, the reversed hazard rate function by
$\tilde r_W(\!\cdot\!)$
, and the survival/reliability function by
$\skew5\bar{F}_W(\!\cdot\!)$
.
Consider a coherent system with lifetime
$\tau (X)$
formed by n dependent components with the lifetime vector
${X}=(X_1,X_2,\ldots,X_n)$
. The dependence among components can be represented by the joint reliability function of
$\textit{X}$
,
$$
\skew5\bar{F}_\textbf{X}(x_1,x_2,\ldots,x_n) =\mathbb{P} (X_1\gt{x_1},X_2\gt{x_2},\ldots,X_n\gt{x_n})
=C (\skew5\bar{F}_{X_1}(x_1),\skew5\bar{F}_{X_2}(x_2),\ldots,\skew5\bar{F}_{X_n}(x_n)),
where
$C(\cdot,\cdot,\ldots,\cdot)$
is a survival copula. In the literature, many different types of survival copulas have been studied, for example, the Farlie–Gumbel–Morgenstern (FGM) copula, the Archimedean copula, the Clayton–Oakes (CO) copula, etc. For more discussion on this, we refer the reader to [Reference Nelsen29]. Based on the above representation, the following fundamental lemma (see [Reference Navarro, Águila, Sordo and Suárez-Liorens26]) can be formulated.
Lemma 1.1. Let
$\tau ({X})$
be the lifetime of a coherent system formed by n dependent components with the lifetime vector
${X}=(X_1,X_2,\ldots,X_n)$
. Then the system’s reliability function can be written as
\[
\skew5\bar{F}_{\tau (\textbf{\textit{X}}) }(x)=h (\skew5\bar{F}_{X_1}(x),\skew5\bar{F}_{X_2}(x),\ldots,\skew5\bar{F}_{X_n}(x)) ,
\]
where
$h\colon [0,1]^n\rightarrow [0,1]$
, called the domination (or dual distortion) function, depends on the structure function
$\phi(\!\cdot\!)$
(see [Reference Barlow and Proschan5] for a definition) and on the survival copula C of
$X_1,X_2,\ldots,X_n$
. Furthermore,
$h(\!\cdot\!)$
is an increasing continuous function in
$[0,1]^n$
such that
$h(0,0,\ldots,0)=0$
and
$h(1,1,\ldots,1)=1$
The following example (borrowed from [Reference Navarro, Águila, Sordo and Suárez-Liorens24]) illustrates the meaning of this lemma.
Example 1.1. Let us consider a coherent system with lifetime
$\tau (\textbf{\textit{X}}) =\min\{X_1,$
$\max\{X_2,X_3\}\}$
, where
$\textbf{\textit{X}}=(X_1,X_2,X_3)$
is described by the FGM copula (see [Reference Nelsen29]). Then the minimal path sets (see [Reference Barlow and Proschan5]) of this system are given by
$P_1=\{1,2\}$
and
$P_2=\{1,3\}$
. Consequently, its reliability function can be obtained as
\begin{align*}
\skew5\bar{F}_{\tau (\textbf{\textit{X}}) }(x)
&= \mathbb{P} (\{X_{P_1}>x\}{\cup}\{X_{P_2}\,>\,x\})\\*
&= \mathbb{P} (X_{P_1}>x)+\mathbb{P}(X_{P_2}>x) -\mathbb{P}(X_{\{1,2,3\}}>x) \\
&= \skew5\bar{F}_\textbf{\textit{X}}(x,x,0)+\skew5\bar{F}_\textbf{\textit{X}}(x,0,x)-\skew5\bar{F}_\textbf{\textit{X}}(x,x,x) \\
&= C (\skew5\bar{F}_{X_1}(x),\skew5\bar{F}_{X_2}(x),1) +C (\skew5\bar{F}_{X_1}(x),1,\skew5\bar{F}_{X_3}(x))
+C (\skew5\bar{F}_{X_1}(x),\skew5\bar{F}_{X_2}(x),\skew5\bar{F}_{X_3}(x)) \\*
&= h (\skew5\bar{F}_{X_1}(x),\skew5\bar{F}_{X_2}(x),\skew5\bar{F}_{X_3}(x)) ,
\end{align*}
where
\begin{align*}
h(\kern1ptp_1,p_2,p_3)&= C(\kern1ptp_1,p_2,1)+C(\kern1ptp_1,1,p_3)-C(\kern1ptp_1,p_2,p_3)
\\*
&= p_1p_2p_3[1-x(1-p_1)(1-p_2)(1-p_3)]\quad \text{for $p_i\in(0,1)$ and $x\in[-1,1]$}.
\end{align*}
Let
$\tau (\textbf{\textit{X}}(\Theta)) $
be a random variable representing the lifetime of a coherent system
$\tau (\textbf{\textit{X}}) $
that is operating in a random environment modelled by a random variable
$\Theta$
with support
$\Omega\subseteq [0,\infty).$
For a given environment
$\Theta=\theta$
, let
$\textbf{\textit{X}}(\theta)=(X_1(\theta),X_2(\theta),\ldots,X_n(\theta))$
be the vector of lifetimes of components, and let
$h(\!\cdot\!)$
be the domination function of
$\tau (\textbf{\textit{X}}(\theta)) $
. Further, let
$\skew5\bar{F}_{X_i}(\cdot\!{\mid}\!\theta)$
,
$ F_{X_i}(\cdot\!{\mid}\!\theta)$
,
$f_{X_i}(\cdot\!{\mid}\!\theta)$
,
$r_{X_i}(\cdot\!{\mid}\!\theta)$
, and
$\tilde r_{X_i}(\cdot\!{\mid}\!\theta)$
be the survival function, the cumulative distribution function, the probability density function, the hazard rate function, and the reversed hazard rate function describing
$X_i(\theta)$
, respectively, for
$i=1,2,\ldots,n.$
Then the reliability function describing
$\tau (\textbf{\textit{X}}(\Theta)) $
can be expressed as the following mixture:
\begin{equation}
\skew5\bar{F}_{\tau (\textbf{\textit{X}}(\Theta)) }(x) = \mathbb{P}(\tau (\textbf{\textit{X}}(\Theta)) >x)
= \int_\Omega h (\skew5\bar{F}_{X_1}(x\!{\mid}\!\theta),\skew5\bar{F}_{X_{2}}(x\!{\mid}\!\theta),\ldots,\skew5\bar{F}_{X_{n}}(x\!{\mid}\!\theta)) \,{\textrm{d}} F _\Theta(\theta),
\label{eqn1}
\end{equation}
where the last equality holds due to Lemma 1.1 Further, its cumulative distribution function is given by
\begin{align}
F_{\tau (\textbf{\textit{X}}(\Theta)) }(x)=\int_\Omega [1-h (\skew5\bar{F}_{X_1}(x\!{\mid}\!\theta),\skew5\bar{F}_{X_2}(x\!{\mid}\!\theta),\ldots,\skew5\bar{F}_{X_n}(x\!{\mid}\!\theta)) ]\,{\textrm{d}} F _\Theta(\theta).
\label{eqn2}
\end{align}
If all the
$X_i$
are identical, then (1.1) and (1.2) reduce to
$$
\skew5\bar{F}_{\tau (\textbf{\textit{X}}(\Theta)) }(x) = \int_\Omega h (\skew5\bar{F}_{X_1}(x\!{\mid}\!\theta)) \,{\textrm{d}} F _\Theta(\theta),
$$
and
$$
F_{\tau (\textbf{\textit{X}}(\Theta)) }(x)=\int_\Omega [1-h (\skew5\bar{F}_{X_1}(x\!{\mid}\!\theta)) ]\,{\textrm{d}} F _\Theta(\theta).
$$
Stochastic orders are frequently used as an effective tool to compare the lifetimes of two systems. Numerous stochastic orders are reported in the literature. Each of them has its own merit. For example, the usual stochastic order compares two reliability functions, the hazard rate order compares two hazard/failure rate functions, whereas the reversed hazard rate order compares two reversed hazard rate functions. For exhaustive details see [Reference Shaked and Shanthikumar37]. For the sake of completeness, we define below the stochastic orders that are used in our paper.
Definition 1.1. Let X and Y be two absolutely continuous random variables with respective supports
$(l_X,u_X)$
and
$(l_Y,u_Y)$
, where
$u_X$
and
$u_Y$
may be positive infinity, and
$l_X$
and
$l_Y$
may be negative infinity. Then, X is said to be smaller than Y in:
(a) the likelihood ratio (
$\text{lr}$
) order, denoted by
$X\leq_{\text{lr}}Y$
, if
\[{f_Y(t)}/{f_X(t)} \text{ is increasing in $ t\in(l_X,u_X)\cup(l_Y,u_Y)$;}\]
(b) the hazard rate (
$\text{hr}$
) order, denoted by
$X\leq_{\text{hr}}Y$
, if
\[{\skew5\bar{F}_Y(t)}/{\skew5\bar{F}_X(t)} \text{ is increasing in $ t \in (-\infty,\max(u_X,u_Y))$;}\]
(c) the reversed hazard rate (
$\text{rhr}$
) order, denoted by
$X\leq_{\text{rhr}}Y$
, if
\[ {F_Y(t)}/{ F_X(t)} \text{ is increasing in $ t \in(\min(l_X,l_Y),\infty)$;}\]
(d) the usual stochastic (
$\text{st}$
) order, denoted by
$X\leq_{\text{st}}Y$
, if
\[\skew5\bar{F}_X(t)\leq \skew5\bar{F}_Y(t) \quad \text{for all }t\in~(-\infty,\infty).\]
Note that the following chain of implications holds among the stochastic orders discussed above:
$$
[X{{\le }_{\text{lr}}}Y\ \Rightarrow X{{\le }_{\text{hr}[\text{rhr}]}}Y\Rightarrow X{{\le }_{\text{st}}}Y.
$$
The theory of totally positive functions has many applications in different areas of approximation theory and related fields. An encyclopedic information on this topic could be found in [Reference Karlin15]. Below we give the definitions of
$\text{TP}_2$
and
$\text{RR}_2$
functions which will be used in our paper.
Definition 1.2. Let
$\mathcal{X}$
and
$\mathcal{Y}$
be two linearly ordered sets. Then, a real-valued non-negative function
$\kappa(\cdot,\cdot)$
defined on
$\mathcal{X}\times\mathcal{Y}$
is said to be
$\text{TP}_2$
(resp.
$\text{RR}_2$
) if
\[
\kappa(x_{1},y_{1})\,\kappa(x_{2},y_{2})\geq(\text{resp.}\leq)\;\kappa(x_{1},y_{2})\,\kappa(x_{2},y_{1})
\]
for all
$x_1<x_2$
and
$y_1<y_2$
.
Throughout the paper, increasing and decreasing, as usual, mean non-decreasing and non-increasing, respectively. All random variables considered in this paper are assumed to be absolutely continuous and non-negative. We use bold symbols to represent vectors. Further, we use the acronym ‘i.i.d.’ for ‘independent and identically distributed’. We let
$\tau_{k:n}$
and
$\tau_{l:m}$
denote the lifetimes of k-out-of-n and l-out-of-m systems, respectively.
The rest of the paper is organized as follows. In Section 2 we formulate some useful lemmas which are used in proving the main results. In Section 3 we consider two coherent systems that operate in the same random environment. We provide some sufficient conditions for proving that one coherent system dominates the other with respect to different stochastic orders. In Section 4 we study the same kind of comparisons under the assumption that different coherent systems operate in different random environments. In Section 5 we assume that one of the coherent systems operates in a random environment and the other in a deterministic environment. Concluding remarks are given in Section 6.
To enhance the readability of the paper, all proofs of theorems, wherever given, are deferred to the Appendix.
2. Useful lemmas
In this section we provide some lemmas which will be used intensively for proving the main results. The first lemma describes the
$\text{TP}_2$
/
$\text{RR}_2$
property of the integral of a function. The proof could be completed along the same lines as in [Reference Dewan and Khaledi9, Lemma 2.1].
Lemma 2.1. Let
$\phi_i(x, \theta)$
,
$i = 1, 2$
, be a non-negative real-valued function on
$\mathbb{R} \times \mathbb{X}$
, where
$\mathbb{R}$
is the set of real numbers, and
$\mathbb{X}\subseteq \mathbb{R}$
. Suppose that the following conditions hold.
(i) For
$\theta \in \mathbb{X}$
,
$\phi_2(x, \theta)/\phi_1(x, \theta)$
is [increasing, increasing, decreasing, decreasing, resp.] in
$x\in\mathbb{R}$
.(ii) For
$x \in \mathbb{R}$
,
$\phi_2(x, \theta)/\phi_1(x, \theta)$
is [increasing, decreasing, decreasing, increasing, resp.] in
$\theta\in \mathbb{X}$
.(iii) Either
$\phi_1(x, \theta)$
or
$\phi_2(x, \theta)$
is [
$\!\text{TP}_2$
,
$\text{RR}_2$
,
$\text{TP}_2$
,
$\text{RR}_2$
, resp.] in
$(x, \theta)\in\mathbb{R} \times \mathbb{X}$
.
Then
\[s_i(x) =\int_{X}\phi_i(x, \theta)w(\theta)\,{\textrm{d}} \theta \text{ is }
[\!\text{TP}_2,\text{TP}_2,\text{RR}_2,\text{RR}_2, \text{resp.}]\ \text{in }(x, i)\in \mathbb{R}\times\{1,2\},\]
where
$w(\!\cdot\!)$
is a continuous function with
$\int_{\mathbb{X}} w(\theta)\,{\textrm{d}} \theta <\infty$
.
In next four lemmas we discuss some properties of the reliability functions of the k-out-of-n and l-out-of-m systems. Lemmas 2.2(i) and 2.4(i) are obtained in [Reference Esary and Proschan10], whereas Lemmas 2.2(ii) and 2.4(ii) are obtained in [Reference Nanda, Jain and Singh23]. Further, Lemmas 2.2(iii), 2.3, 2.5(i), and 2.6 are obtained in [Reference Belzunce, Franco, Ruiz and Ruiz6]. Furthermore, Lemmas 2.5(ii) and 2.5(iv) are obtained in [Reference Hazra and Nanda14], whereas Lemmas 2.5(iii) and 2.5(v) could be proved along the same lines as Lemmas 5 and 7 of [Reference Hazra and Nanda14].
Lemma 2.2. Let
$h_{k:n}(\!\cdot\!)$
be the reliability function of the k-out-of-n system with i.i.d. components. Then the following results hold.
(i)
$\dfrac{ph'_{k:n}(\kern1ptp)}{h_{k:n}(\kern1ptp)}$
is decreasing in
$p\in(0,1)$
.(ii)
$\dfrac{(1-p)h'_{k:n}(\kern1ptp)}{1-h_{k:n}(\kern1ptp)}$
is increasing in
$p\in(0,1)$
.(iii) There exists some point
$\mu \in (0,1)$
such that
(a)
$\dfrac{ph''_{k:n}(\kern1ptp)}{h'_{k:n}(\kern1ptp)}$
is decreasing and positive for all
$ p \in(0,\mu),$
and(b)
$\dfrac{(1-p)h''_{k:n}(\kern1ptp)}{h'_{k:n}(\kern1ptp)}$
is decreasing and negative for all
$ p \in(\mu,1),$
where
$\mu=(k-1)/(n-1)$
.
Lemma 2.3. Let
$h_{k:n}(\!\cdot\!)$
and
$h_{l:m}(\!\cdot\!)$
be the reliability functions of the k-out-of-n and the l-out-of-m systems with i.i.d. components, respectively. Then, for
$l\leq k$
and
$n-k\leq m-l$
,
(i)
$h_{k:n}(\kern1ptp)\leq h_{l:m}(\kern1ptp)$
for all
$p\in(0,1)$
,(ii)
$\dfrac{h_{k:n}(\kern1ptp)}{h_{l:m}(\kern1ptp)}$
is increasing
$p\in(0,1)$
,(iii)
$\dfrac{1-h_{k:n}(\kern1ptp)}{1-h_{l:m}(\kern1ptp)}$
is increasing
$p\in(0,1)$
,(iv)
$\dfrac{h_{k:n}'(\kern1ptp)}{h'_{l:m}(\kern1ptp)} $
is increasing
$p\in(0,1)$
.
Lemma 2.4. Let
$h_{k:n}(\!\cdot\!)$
be the reliability function of the k-out-of-n system with independent components. Then
(i)
$\displaystyle\sum_{i=1}^n\dfrac{p_i}{h_{k:n} (\textbf{\textit{\:p}}) }\dfrac{\partial h_{k:n} (\textbf{\textit{\:p}}) }{\partial p_i}$
is decreasing in each
$p_i\in(0,1)$
for all
$i=1,2,\ldots,n$
,(ii)
$\displaystyle\sum_{i=1}^n\dfrac{1-p_i}{1-h_{k:n} (\textbf{\textit{\:p}}) }\dfrac{\partial h_{k:n} (\textbf{\textit{\:p}}) }{\partial p_i}$
is increasing in each
$p_i\in(0,1)$
for all
$i=1,2,\ldots,n$
.
Lemma 2.5. Let
$h_{k:n}(\!\cdot\!)$
and
$h_{l:m}(\!\cdot\!)$
be the reliability functions of the k-out-of-n and the l-out-of-m systems with independent components, respectively. Then, for
$l\leq k$
and
$n-k\leq m-l$
,
(i)
$h_{k:n} (\textbf{\textit{\:p}}) \leq h_{l:m} (\textbf{\textit{\:p}}) $
,(ii)
$\dfrac{1}{h_{k:n} (\textbf{\textit{\:p}}) }\dfrac{\partial h_{k:n} (\textbf{\textit{\:p}}) }{\partial p_i}\geq \dfrac{1}{h_{l:m} (\textbf{\textit{\:p}}) }\dfrac{\partial h_{l:m} (\textbf{\textit{\:p}}) }{\partial p_i}$
for all
$i=1,2,\ldots,\min\{m,n\}$
,(iii)
$\dfrac{1}{1-h_{k:n} (\textbf{\textit{\:p}}) }\dfrac{\partial h_{k:n} (\textbf{\textit{\:p}}) }{\partial p_i}\leq \dfrac{1}{1-h_{l:m} (\textbf{\textit{\:p}}) }\dfrac{\partial h_{l:m} (\textbf{\textit{\:p}}) }{\partial p_i}$
for all
$i=1,2,\ldots,\min\{m,n\}$
,(iv)
$\displaystyle\sum_{i=1}^n\dfrac{p_i}{h_{k:n} (\textbf{\textit{\:p}}) }\dfrac{\partial h_{k:n} (\textbf{\textit{\:p}}) }{\partial p_i}\geq \sum_{i=1}^m\dfrac{p_i}{h_{l:m} (\textbf{\textit{\:p}}) }\dfrac{\partial h_{l:m} (\textbf{\textit{\:p}}) }{\partial p_i}$
,(v)
$\displaystyle\sum_{i=1}^n\dfrac{1-p_i}{1-h_{k:n} (\textbf{\textit{\:p}}) }\dfrac{\partial h_{k:n} (\textbf{\textit{\:p}}) }{\partial p_i}\leq \sum_{i=1}^m\dfrac{1-p_i}{1-h_{l:m} (\textbf{\textit{\:p}}) }\dfrac{\partial h_{l:m} (\textbf{\textit{\:p}}) }{\partial p_i}.$
Lemma 2.6. Let
$h_{k:n}(\!\cdot\!)$
and
$h_{l:m}(\!\cdot\!)$
be the reliability functions of the k-out-of-n and the l-out-of-m systems, respectively. Further, let
\[\textbf{\textit{Z}}=(Z_1,Z_2,\ldots,Z_n)\quad\text{and}\quad \textbf{\textit{W}}=(W_1,W_2,\ldots,W_m)\]
be two sets of independent component lifetimes. Suppose that, for all
$i=1,2,\ldots,n$
and
$j=1,2,\ldots,m$
,
$Z_i\leq_{\text{lr}}W_j$
. Then, for
$l\leq k$
and
$n-k\leq m-l$
,
\[\dfrac{\partial h_{l:m}(\textbf{\textit{q}})}{\partial q_j}\bigg/\dfrac{\partial h_{k:n}(\textbf{\textit{\:p}})}{\partial p_i} \text{ is increasing in \textit{x},}\]
where
$p_i=\skew5\bar{F}_{Z_i}(x)$
and
$q_j=\skew5\bar{F}_{W_j}(x)$
.
3. Two different coherent systems in the same random environment
In this section we consider two coherent systems with lifetimes
$\tau_1 (\textbf{\textit{X}}(\Theta)) $
and
$\tau_2 (\textbf{\textit{Y}}(\Theta)) $
that operate in the same random environment described by a random variable
$\Theta$
with support
$\Omega$
. For a given (realization) environment
$\Theta=\theta$
, we denote the domination functions of
$\tau_1 (\textbf{\textit{X}}(\theta)) $
and
$\tau_2 (\textbf{\textit{Y}}(\theta)) $
by
$h_1(\!\cdot\!)$
and
$h_2(\!\cdot\!)$
, respectively. In what follows, we provide some sufficient conditions for proving that one coherent system dominates the other with respect to different stochastic orders.
3.1. Systems with not necessarily identical components
In this subsection we consider coherent systems that are formed by not necessarily identical components.
In the following theorem, whose proof is deferred to the Appendix, we compare two coherent systems with respect to the usual stochastic order.
Theorem 3.1. Let
$\textbf{\textit{X}}=(X_1,X_2,\ldots,X_{n})$
and
$\textbf{\textit{Y}}=(Y_1,Y_2,\ldots,Y_{m})$
be two sets of components’ lifetimes. Suppose that the following conditions hold:
(i)
$h_1(\kern1ptp_1,p_2,\ldots,p_n)\leq h_2(\kern1ptp_1,p_2,\ldots,p_m)$
,(ii)
$X_i(\theta)\leq_{\text{st}}Y_i(\theta)$
for all
$i=1,2,\ldots,\min\{m,n\}.$
Then
$\tau_1 (\textbf{\textit{X}}(\Theta)) \leq_{\text{st}}\tau_2 (\textbf{\textit{Y}}(\Theta)) $
.
In the next theorem (see the Appendix for the proof), we show that, under some sufficient conditions,
$\tau_2 (\textbf{\textit{Y}}(\Theta)) $
is larger than
$\tau_1 (\textbf{\textit{X}}(\Theta)) $
with respect to the hazard rate order.
Theorem 3.2. Let
$\textbf{\textit{X}}=(X_1,X_2,\ldots,X_{n})$
and
$\textbf{\textit{Y}}=(Y_1,Y_2,\ldots,Y_{m})$
be two sets of components’ lifetimes, where
$n\geq m$
. Suppose that
$\{\mathrm{(i)},\mathrm{(ii)},\mathrm{(iii)}\}$
or
$\{\mathrm{(i)},\mathrm{(ii)},\mathrm{(iv)}\}$
holds:
(i)
$\dfrac{1}{h_1 (\textbf{\textit{\:p}}) }\dfrac{\partial h_1 (\textbf{\textit{\:p}}) }{\partial p_i}\geq\dfrac{1}{{h_2} (\textbf{\textit{\:p}}) }\dfrac{\partial {h_2} (\textbf{\textit{\:p}}) }{\partial p_i}$
for all
$i=1,2,\ldots,m$
,(ii)
$\dfrac{p_i}{h_2 (\textbf{\textit{\:p}}) }\dfrac{\partial h_2 (\textbf{\textit{\:p}}) }{\partial p_i}$
is decreasing in
$\textbf{\textit{\:p}}\in(0,1)^m$
for all
$i=1,2,\ldots,m$
,(iii)
$X_i(\theta_1)\leq_{\text{hr}}X_i(\theta_2),\;\;X_j(\theta)\leq_{\text{hr}}Y_j(\theta), \text{ and }Y_j(\theta_2)\leq_{\text{hr}}Y_j(\theta_1)$
for all
$\theta, \theta_1, \theta_2\in~\Omega$
such that
$\theta_1\leq \theta_2,$
and for all
$i=1,2,\ldots,n$
and
$j=1,2,\ldots,m$
,(iv)
$X_i(\theta_1)\geq_{\text{hr}}X_i(\theta_2),\;\;X_j(\theta)\leq_{\text{hr}}Y_j(\theta), \text{ and }Y_j(\theta_2)\geq_{\text{hr}}Y_j(\theta_1) $
for all
$\theta, \theta_1, \theta_2\in~\Omega$
such that
$\theta_1\leq \theta_2,$
and for all
$i=1,2,\ldots,n$
and
$j=1,2,\ldots,m$
.
Then
$\tau_1 (\textbf{\textit{X}}(\Theta)) \leq_{\text{hr}}\tau_2 (\textbf{\textit{Y}}(\Theta)) $
.
The following theorem shows that the above result holds for the reversed hazard rate order under some different set of sufficient conditions. The proof could be completed along the same lines as in Theorem 3.2 and is hence omitted.
Theorem 3.3. Let
$\textbf{\textit{X}}=(X_1,X_2,\ldots,X_{n})$
and
$\textbf{\textit{Y}}=(Y_1,Y_2,\ldots,Y_{m})$
be two sets of components’ lifetimes, where
$m\geq n$
. Suppose that
$\{\mathrm{(i)},\mathrm{(ii)},\mathrm{(iii)}\}$
or
$\{\mathrm{(i)},\mathrm{(ii)},\mathrm{(iv)}\}$
holds:
(i)
$\dfrac{1}{1-h_1 (\textbf{\textit{\:p}}) }\dfrac{\partial h_1 (\textbf{\textit{\:p}}) }{\partial p_i}\leq\dfrac{1}{{1-h_2} (\textbf{\textit{\:p}}) }\dfrac{\partial {h_2} (\textbf{\textit{\:p}}) }{\partial p_i}$
for all
$i=1,2,\ldots,n$
,(ii)
$\dfrac{1-p_i}{1-h_1 (\textbf{\textit{\:p}}) }\dfrac{\partial h_1 (\textbf{\textit{\:p}}) }{\partial p_i}$
is increasing in
$\textbf{\textit{\:p}}\in(0,1)^n$
for all
$i=1,2,\ldots,n$
,(iii)
$X_i(\theta_1)\leq_{\text{rhr}}X_i(\theta_2),\ X_i(\theta)\leq_{\text{rhr}}Y_i(\theta), \text{ and }Y_j(\theta_2)\leq_{\text{rhr}}Y_j(\theta_1)$
for all
$\theta, \theta_1, \theta_2\in \Omega$
such that
$\theta_1\leq~\theta_2,$
and for all
$i=1,2,\ldots,n$
and
$j=1,2,\ldots,m$
,(iv)
$X_i(\theta_1)\geq_{\text{rhr}}X_i(\theta_2),\;\;X_i(\theta)\leq_{\text{rhr}}Y_i(\theta), \text{ and }Y_j(\theta_2)\geq_{\text{rhr}}Y_j(\theta_1)$
for all
$\theta, \theta_1, \theta_2\in~\Omega$
such that
$\theta_1\leq~\theta_2,$
and for all
$i=1,2,\ldots,n$
and
$j=1,2,\ldots,m$
.
Then
$\tau_1 (\textbf{\textit{X}}(\Theta)) \leq_{\text{rhr}}\tau_2 (\textbf{\textit{Y}}(\Theta)) $
.
3.2. Systems with identical components
In this subsection we consider coherent systems of identical components. Obviously, this case has its own value when compared with the general case of non-identical components. The following theorem is analogous to Theorem 3.1, and the proof also immediately follows from it.
Theorem 3.4. Let
$\textbf{\textit{X}}=(X_1,X_2,\ldots,X_{n})$
and
$\textbf{\textit{Y}}=(Y_1,Y_2,\ldots,Y_{m})$
be two sets of components’ lifetimes. Assume the
$X_i$
are identical and the
$Y_j$
are identical. Suppose that the following conditions hold:
(i)
$h_1(\kern1ptp)\leq h_2(\kern1ptp)$
for all
$p\in (0,1)$
,(ii)
$X_1(\theta)\leq_{\text{st}}Y_1(\theta)$
for all
$\theta\in \Omega$
.
Then
$\tau_1 (\textbf{\textit{X}}(\Theta)) \leq_{\text{st}}\tau_2 (\textbf{\textit{Y}}(\Theta)) $
.
In the next theorem we compare two coherent systems with respect to the hazard rate order. The proof could be completed along the same lines as in Theorem 3.2.
Theorem 3.5. Let
$\textbf{\textit{X}}=(X_1,X_2,\ldots,X_{n})$
and
$\textbf{\textit{Y}}=(Y_1,Y_2,\ldots,Y_{m})$
be two sets of components’ lifetimes. Assume the
$X_i$
are identical and the
$Y_j$
are identical. Suppose that
$\{\mathrm{(i)},\mathrm{(ii)},\mathrm{(iii)}\}$
or
$\{\mathrm{(i)},\mathrm{(ii)},\mathrm{(iv)}\}$
holds:
(i)
$h_1(\kern1ptp)/ h_2(\kern1ptp)$
is increasing in
$p\in (0,1)$
,(ii)
$ph_2'(\kern1ptp)/h_2(\kern1ptp)$
is decreasing in
$p\in (0,1)$
,(iii)
$X_1(\theta_1)\leq_{\text{hr}}X_1(\theta_2)\leq_{\text{hr}}Y_1(\theta_2)\leq_{\text{hr}}Y_1(\theta_1)$
for all
$\theta_1, \theta_2\in \Omega\text{ such that }\theta_1\leq ~\theta_2$
,(iv)
$X_1(\theta_2)\leq_{\text{hr}}X_1(\theta_1)\leq_{\text{hr}}Y_1(\theta_1)\leq_{\text{hr}}Y_1(\theta_2)$
for all
$\theta_1, \theta_2\in \Omega\text{ such that }\theta_1\leq ~\theta_2.$
Then
$\tau_1 (\textbf{\textit{X}}(\Theta)) \leq_{\text{hr}}\tau_2 (\textbf{\textit{Y}}(\Theta)) $
.
In the following theorem we show that the same result as in Theorem 3.5 holds for the reversed hazard rate order. The proof is similar to that of Theorem 3.3.
Theorem 3.6. Let
$\textbf{\textit{X}}=(X_1,X_2,\ldots,X_{n})$
and
$\textbf{\textit{Y}}=(Y_1,Y_2,\ldots,Y_{m})$
be two sets of components’ lifetimes. Assume the
$X_i$
are identical and the
$Y_j$
are identical. Suppose that
$\{\mathrm{(i)},\mathrm{(ii)},\mathrm{(iii)}\}$
or
$\{\mathrm{(i)},\mathrm{(ii)},\mathrm{(iv)}\}$
holds:
(i)
$(1-h_1(\kern1ptp))/(1- h_2(\kern1ptp))$
is increasing in
$p\in (0,1)$
,(ii)
$(1-p)h_1'(\kern1ptp)/(1-h_1(\kern1ptp))$
is increasing in
$p\in (0,1)$
,(iii)
$X_1(\theta_1)\leq_{\text{rhr}}X_1(\theta_2)\leq_{\text{rhr}}Y_1(\theta_2)\leq_{\text{rhr}}Y_1(\theta_1)$
for all
$\theta_1, \theta_2\in \Omega\text{ such that }\theta_1\leq~ \theta_2$
,(iv)
$X_1(\theta_2)\leq_{\text{rhr}}X_1(\theta_1)\leq_{\text{rhr}}Y_1(\theta_1)\leq_{\text{rhr}}Y_1(\theta_2)$
for all
$\theta_1, \theta_2\in \Omega\text{ such that }\theta_1\leq ~\theta_2$
.
Then
$\tau_1 (\textbf{\textit{X}}(\Theta)) \leq_{\text{rhr}}\tau_2 (\textbf{\textit{Y}}(\Theta)) $
.
4. Different coherent systems in different random environments
In this section we consider two coherent systems with lifetimes
$\tau_1 (\textbf{\textit{X}}(\Theta_1)) $
and
$\tau_2 (\textbf{\textit{Y}}(\Theta_2)) $
, where
$\Theta_1$
and
$\Theta_2$
are two random variables (with support
$\Omega$
) that describe two different random environments. For given environments
$\Theta_1=\theta$
and
$\Theta_2=\theta^*$
, we denote the domination functions of
$\tau_1 (\textbf{\textit{X}}(\theta)) $
and
$\tau_2 (\textbf{\textit{Y}}(\theta^*)) $
by
$h_1(\!\cdot\!)$
and
$h_2(\!\cdot\!)$
, respectively. We will compare
$\tau_1 (\textbf{\textit{X}}(\Theta_1)) $
and
$\tau_2 (\textbf{\textit{Y}}(\Theta_2)) $
with respect to different stochastic orders. It should be noted that the results of this section can be considered as generalizations of the corresponding results given in [Reference Amini-Seresht, Zhang and Balakrishnan1] to the case when there are two different coherent systems; in [Reference Amini-Seresht, Zhang and Balakrishnan1], the case of one system (or of two identical systems) operating in two environments was discussed.
4.1. Systems with not necessarily identical components
In this subsection we assume that coherent systems are formed by not necessarily identical components.
In the following theorem, we show that under a set of sufficient conditions
$\tau_2 (\textbf{\textit{Y}}(\Theta_2)) $
dominates
$\tau_1 (\textbf{\textit{X}}(\Theta_1)) $
with respect to the usual stochastic order. The proof follows from [1, Theorem 3.1] and our Theorem 3.1.
Theorem 4.1. Let
$\textbf{\textit{X}}=(X_1,X_2,\ldots,X_{n})$
and
$\textbf{\textit{Y}}=(Y_1,Y_2,\ldots,Y_{m})$
be two sets of components’ lifetimes. Suppose that
$\{\mathrm{(i)},\mathrm{(ii)},\mathrm{(iv)}\}$
or
$\{\mathrm{(i)},\mathrm{(iii)},\mathrm{(iv)}\}$
holds:
(i)
$h_1(\kern1ptp_1,p_2,\ldots,p_n)\leq h_2(\kern1ptp_1,p_2,\ldots,p_m)$
,(ii)
$X_i(\theta_1)\leq_{\text{st}}X_i(\theta_2)$
and
$X_j(\theta)\leq_{\text{st}}Y_j(\theta)$
for all
$\theta, \theta_1, \theta_2\in \Omega$
such that
$\theta_1\leq \theta_2,$
and for all
$i=1,2,\ldots,n$
and
$j=1,2,\ldots,\min\{m,n\}$
,(iii)
$Y_i(\theta_1)\leq_{\text{st}}Y_i(\theta_2)$
and
$X_j(\theta)\leq_{\text{st}}Y_j(\theta)$
for all
$\theta, \theta_1, \theta_2\in \Omega$
such that
$\theta_1\leq \theta_2,$
and for all
$i=1,2,\ldots,m$
and
$j=1,2,\ldots,\min\{m,n\}$
,(iv)
$\Theta_1\leq_{\text{st}} \Theta_2$
.
Then
$\tau_1 (\textbf{\textit{X}}(\Theta_1)) \leq_{\text{st}}\tau_2 (\textbf{\textit{Y}}(\Theta_2)) $
.
Now we compare two coherent systems with respect to the hazard rate order. The proof follows from [Reference Amini-Seresht, Zhang and Balakrishnan1, Theorem 3.2] and our Theorem 3.2.
Theorem 4.2. Let
$\textbf{\textit{X}}=(X_1,X_2,\ldots,X_{n})$
and
$\textbf{\textit{Y}}=(Y_1,Y_2,\ldots,Y_{m})$
be two sets of components’ lifetimes, where
$n\geq m$
. Suppose that
$\{\mathrm{(i)},\mathrm{(ii)},\mathrm{(iii)},\mathrm{(v)}\}$
or
$\{\mathrm{(i)},\mathrm{(ii)},\mathrm{(iv)},\mathrm{(v)}\}$
holds:
(i)
$\dfrac{1}{h_1 (\textbf{\textit{\:p}}) }\dfrac{\partial h_1 (\textbf{\textit{\:p}}) }{\partial p_i}\geq\dfrac{1}{{h_2} (\textbf{\textit{\:p}}) }\dfrac{\partial {h_2} (\textbf{\textit{\:p}}) }{\partial p_i}$
for all
$i=1,2,\ldots,m$
,(ii)
$\dfrac{p_i}{h_2 (\textbf{\textit{\:p}}) }\dfrac{\partial h_2 (\textbf{\textit{\:p}}) }{\partial p_i}$
is decreasing in
$\textbf{\textit{\:p}}\in(0,1)^m$
for all
$i=1,2,\ldots,m$
,(iii)
$X_i(\theta_1)\leq_{\text{hr}}X_i(\theta_2),\;\;X_j(\theta)\leq_{\text{hr}}Y_j(\theta), \text{ and }Y_j(\theta_2)\leq_{\text{hr}}Y_j(\theta_1)$
for all
$\theta, \theta_1, \theta_2\in~\Omega$
such that
$\theta_1\leq \theta_2,$
and for all
$i=1,2,\ldots,n$
and
$j=1,2,\ldots,m$
,(iv)
$X_i(\theta_1)\geq_{\text{hr}}X_i(\theta_2),\;\;X_j(\theta)\leq_{\text{hr}}Y_j(\theta), \text{ and }Y_j(\theta_2)\geq_{\text{hr}}Y_j(\theta_1)$
for all
$\theta, \theta_1, \theta_2\in~\Omega$
such that
$\theta_1\leq \theta_2,$
and for all
$i=1,2,\ldots,n$
and
$j=1,2,\ldots,m$
,(v)
$\Theta_1\leq_{\text{hr}} \Theta_2$
.
Then
$\tau_1 (\textbf{\textit{X}}(\Theta_1)) \leq_{\text{hr}}\tau_2 (\textbf{\textit{Y}}(\Theta_2)) $
.
Below we give two examples that illustrate conditions (i) and (ii) of Theorem 4.2.
Example 4.1. Consider two coherent systems with lifetimes
$\tau_1(\textbf{\textit{X}})=\min\{X_1,X_2,X_3\}$
and
$\tau_2(\textbf{\textit{Y}})=\min\{X_1,\max\{X_2,X_3\}\}$
. Further, let both
$\{X_1,X_2,X_3\}$
and
$\{Y_1,Y_2,Y_3\}$
have the same Gumbel–Barnett copula given by
\[C(\,p_1,p_2,p_3)=p_1p_2p_3\,{\textrm{e}}^{\alpha \ln p_1\ln p_2 \ln p_3}, \quad \alpha>0,\text{ and }0<p_i<1 \quad \text{for }i=1,2,3.\]
Then the domination functions of
$\tau_1(\textbf{\textit{X}})$
and
$\tau_2(\textbf{\textit{Y}})$
, respectively, are given by
\[h_1(\textbf{\textit{\:p}})=C(\,p_1,p_2,p_3)=p_1p_2p_3\,{\textrm{e}}^{\alpha \ln p_1\ln p_2 \ln p_3}\]
and
\[
h_2(\textbf{\textit{\:p}}) = C(\kern1ptp_1,p_2,1)+C(\kern1ptp_1,1,p_3)-C(\kern1ptp_1,p_2,p_3)
= p_1p_2+p_1p_3-p_1p_2p_3\,{\textrm{e}}^{\alpha \ln p_1\ln p_2 \ln p_3}.
\]
Note that
\begin{align}
\dfrac{p_i}{h_1 (\textbf{\textit{\:p}}) }\dfrac{\partial h_1 (\textbf{\textit{\:p}}) }{\partial p_i}=1+\alpha \prod_{\substack{j= 1\\ j\neq i}}^3 \ln p_j\quad \text{for } i=1,2,3,
\label{eqn3}
\end{align}
and
\begin{align}
\dfrac{p_1}{h_2 (\textbf{\textit{\:p}}) }\dfrac{\partial h_2 (\textbf{\textit{\:p}}) }{\partial p_1} &=1- \dfrac{\alpha p_1p_2p_3 \ln p_2\ln p_3\,{\textrm{e}}^{\alpha \ln p_1\ln p_2 \ln p_3}}{p_1p_2+p_1p_3-p_1p_2p_3\,{\textrm{e}}^{\alpha \ln p_1\ln p_2 \ln p_3}},
\label{eqn4}\end{align}
\begin{align}*
\dfrac{p_2}{h_2 (\textbf{\textit{\:p}}) }\dfrac{\partial h_2 (\textbf{\textit{\:p}}) }{\partial p_2} &=1- \dfrac{p_1p_3+\alpha p_1p_2p_3 \ln p_1\ln p_3\,{\textrm{e}}^{\alpha \ln p_1\ln p_2 \ln p_3}}{p_1p_2+p_1p_3-p_1p_2p_3\,{\textrm{e}}^{\alpha \ln p_1\ln p_2 \ln p_3}},
\label{eqn5}\end{align}
\begin{align}*
\dfrac{p_3}{h_2 (\textbf{\textit{\:p}}) }\dfrac{\partial h_2 (\textbf{\textit{\:p}}) }{\partial p_3} &=1- \dfrac{p_1p_2+\alpha p_1p_2p_3 \ln p_2\ln p_3\,{\textrm{e}}^{\alpha \ln p_1\ln p_2 \ln p_3}}{p_1p_2+p_1p_3-p_1p_2p_3\,{\textrm{e}}^{\alpha \ln p_1\ln p_2 \ln p_3}}.
\label{eqn6}\end{align}
\[\dfrac{1}{h_1 (\textbf{\textit{\:p}}) }\dfrac{\partial h_1 (\textbf{\textit{\:p}}) }{\partial p_i}\geq\dfrac{1}{{h_2} (\textbf{\textit{\:p}}) }\dfrac{\partial {h_2} (\textbf{\textit{\:p}}) }{\partial p_i}\quad\text{for all $ i=1,2,3$.}\]
Example 4.2. Consider two coherent systems
\[
\tau_1(\textbf{\textit{X}})=\min\{X_1,X_2,\ldots,X_n\},\quad
\tau_2(\textbf{\textit{Y}})=\min\{Y_1,Y_2,\ldots,Y_n\},
\]
where n is even. Further, let both
$\{X_1,X_2,\ldots,X_n\}$
and
$\{Y_1,Y_2,\ldots,Y_n\}$
have the same Gumbel–Barnett copula given by
$$
C(\kern1ptp_1,p_2,\ldots,p_n)=\prod_{i=1}^np_i\,{\textrm{e}}^{-\alpha (-1)^n\prod_{j=1}^n\ln p_j}, \quad \alpha\gt0,\text{ and }0\lt{p_i}\lt1 \text{ for } i=1,2,\ldots,n.
$$
Then
$\tau_1(\textbf{\textit{X}})$
and
$\tau_2(\textbf{\textit{Y}})$
have the same domination function:
\[
h_i(\textbf{\textit{\:p}})=C(\kern1ptp_1,p_2,\ldots,p_n) \quad\text{for } i=1,2.
\]
Note that, for all
$i=1,2,\ldots,n$
,
\[\dfrac{p_i}{h_2 (\textbf{\textit{\:p}}) }\dfrac{\partial h_2 (\textbf{\textit{\:p}}) }{\partial p_i}=1-\alpha\prod_{\substack{j= 1\\ j\neq i}}^n \ln p_j,\]
which is decreasing in
$\textbf{\textit{\:p}}\in(0,1)^n.$
In the following theorem we provide a set of sufficient conditions for proving that
$\tau_2 (\textbf{\textit{Y}}(\Theta_2)) $
is larger than
$\tau_1 (\textbf{\textit{X}}(\Theta_1)) $
with respect to the reversed hazard rate order. The proof follows from [Reference Amini-Seresht, Zhang and Balakrishnan1, Theorem 3.3] and Theorem 3.3.
Theorem 4.3. Let
$\textbf{\textit{X}}=(X_1,X_2,\ldots,X_{n})$
and
$\textbf{\textit{Y}}=(Y_1,Y_2,\ldots,Y_{m})$
be two sets of components’ lifetimes, where
$m\geq n$
. Suppose that
$\{\mathrm{(i)},\mathrm{(ii)},\mathrm{(iii)},\mathrm{(v)}\}$
or
$\{\mathrm{(i)},\mathrm{(ii)},\mathrm{(iv)},\mathrm{(v)}\}$
holds:
(i)
$\dfrac{1}{1-h_1 (\textbf{\textit{\:p}}) }\dfrac{\partial h_1 (\textbf{\textit{\:p}}) }{\partial p_i}\leq\dfrac{1}{{1-h_2} (\textbf{\textit{\:p}}) }\dfrac{\partial {h_2} (\textbf{\textit{\:p}}) }{\partial p_i}$
for all
$i=1,2,\ldots,n$
,(ii)
$\dfrac{1-p_i}{1-h_1 (\textbf{\textit{\:p}}) }\dfrac{\partial h_1 (\textbf{\textit{\:p}}) }{\partial p_i}$
is increasing in
$\textbf{\textit{\:p}}\in(0,1)^n$
for all
$i=1,2,\ldots,n$
,(iii)
$X_i(\theta_1)\leq_{\text{rhr}}X_i(\theta_2),\;\;X_i(\theta)\leq_{\text{rhr}}Y_i(\theta), \text{ and }Y_j(\theta_2)\leq_{\text{rhr}}Y_j(\theta_1) $
for all
$\theta, \theta_1, \theta_2\in~\Omega$
such that
$\theta_1\leq~\theta_2,$
and for all
$i=1,2,\ldots,n$
and
$j=1,2,\ldots,m$
,(iv)
$X_i(\theta_1)\geq_{\text{rhr}}X_i(\theta_2),\;\;X_i(\theta)\leq_{\text{rhr}}Y_i(\theta), \text{ and }Y_j(\theta_2)\geq_{\text{rhr}}Y_j(\theta_1) $
for all
$\theta, \theta_1, \theta_2\in~\Omega$
such that
$\theta_1\leq~\theta_2,$
and for all
$i=1,2,\ldots,n$
and
$j=1,2,\ldots,m$
,(v)
$\Theta_1\leq_{\text{rhr}} \Theta_2$
.
Then
$\tau_1 (\textbf{\textit{X}}(\Theta_1)) \leq_{\text{rhr}}\tau_2 (\textbf{\textit{Y}}(\Theta_2)) $
.
Remark 4.1. It is to be noted that condition (i) of Theorems 4.2 and 4.3 holds for
$h_1 (\textbf{\textit{\:p}}) =h_{k:n} (\textbf{\textit{\:p}}) $
and
$h_2 (\textbf{\textit{\:p}}) =h_{l:m} (\textbf{\textit{\:p}}) $
,
$l\leq k$
and
$n-k\leq m-l$
, where
$h_{k:n}(\!\cdot\!)$
and
$h_{l:m}(\!\cdot\!)$
are the reliability functions of the k-out-of-n and the l-out-of-m systems with independent components, respectively (see Lemma 2.5).
4.2. Systems with identical components
In this subsection we compare two coherent systems that are formed by identical components. Obviously, this is of interest when compared with the general case of non-identical components.
We show that under a set of sufficient conditions
$\tau_2 (\textbf{\textit{Y}}(\Theta_2)) $
is larger than
$\tau_1 (\textbf{\textit{X}}(\Theta_1)) $
with respect to the usual stochastic order. The proof follows from [Reference Amini-Seresht, Zhang and Balakrishnan1, Theorem 3.1] and Theorem 3.4.
Theorem 4.4. Let
$\textbf{\textit{X}}=(X_1,X_2,\ldots,X_{n})$
and
$\textbf{\textit{Y}}=(Y_1,Y_2,\ldots,Y_{m})$
be two sets of components’ lifetimes. Assume the
$X_i$
are identical and the
$Y_j$
are identical. Suppose that
$\{\mathrm{(i)},\mathrm{(ii)},\mathrm{(iv)}\}$
or
$\{\mathrm{(i)},\mathrm{(iii)},\mathrm{(iv)}\}$
holds:
(i)
$h_1(\kern1ptp)\leq h_2(\kern1ptp)$
for all
$p\in (0,1)$
,(ii)
$X_1(\theta_1)\leq_{\text{st}}X_1(\theta_2)$
and
$X_1(\theta)\leq_{\text{st}}Y_1(\theta)$
for all
$\theta,\theta_1, \theta_2\in \Omega\text{ such that }\theta_1\leq ~\theta_2$
,(iii)
$Y_1(\theta_1)\leq_{\text{st}}Y_1(\theta_2)$
and
$X_1(\theta)\leq_{\text{st}}Y_1(\theta)$
for all
$\theta,\theta_1, \theta_2\in \Omega\text{ such that }\theta_1\leq ~\theta_2$
,(iv)
$\Theta_1\leq_{\text{st}} \Theta_2$
.
Then
$\tau_1 (\textbf{\textit{X}}(\Theta_1)) \leq_{\text{st}}\tau_2 (\textbf{\textit{Y}}(\Theta_2)) $
.
The next theorem discusses the same result as in the above theorem but with respect to the hazard rate order. The proof follows from [Reference Amini-Seresht, Zhang and Balakrishnan1, Corollary 3.1] and Theorem 3.5.
Theorem 4.5. Let
$\textbf{\textit{X}}=(X_1,X_2,\ldots,X_{n})$
and
$\textbf{\textit{Y}}=(Y_1,Y_2,\ldots,Y_{m})$
be two sets of components’ lifetimes. Assume the
$X_i$
are identical and the
$Y_j$
are identical. Suppose that
$\{\mathrm{(i)},\mathrm{(ii)},\mathrm{(iii)},\mathrm{(v)}\}$
or
$\{\mathrm{(i)},\mathrm{(ii)},\mathrm{(iv)},\mathrm{(v)}\}$
holds:
(i)
$h_1(\kern1ptp)/ h_2(\kern1ptp)$
is increasing in
$p\in (0,1)$
,(ii)
$ph_2'(\kern1ptp)/h_2(\kern1ptp)$
is decreasing in
$p\in (0,1)$
,(iii)
$X_1(\theta_1)\leq_{\text{hr}}X_1(\theta_2)\leq_{\text{hr}}Y_1(\theta_2)\leq_{\text{hr}}Y_1(\theta_1)$
for all
$\theta_1, \theta_2\in \Omega\text{ such that }\theta_1\leq ~\theta_2$
,(iv)
$X_1(\theta_2)\leq_{\text{hr}}X_1(\theta_1)\leq_{\text{hr}}Y_1(\theta_1)\leq_{\text{hr}}Y_1(\theta_2)$
for all
$\theta_1, \theta_2\in \Omega\text{ such that }\theta_1\leq ~\theta_2$
,(v)
$\Theta_1\leq_{\text{hr}} \Theta_2$
.
Then
$\tau_1 (\textbf{\textit{X}}(\Theta_1)) \leq_{\text{hr}}\tau_2 (\textbf{\textit{Y}}(\Theta_2)) $
.
The following example illustrates conditions (i) and (ii) of Theorem 4.5.
Example 4.3. Consider two coherent systems with lifetimes
$\tau_1(\textbf{\textit{X}})=\min\{X_1,X_2,X_3\}$
and
$\tau_2(\textbf{\textit{Y}})=\min\{Y_1,\max\{Y_2,Y_3\}\}$
, where both
$\{X_1,X_2,X_3\}$
and
$\{Y_1,Y_2,Y_3\}$
are homogeneous and have the same FGM copula given by
\[C(\kern1ptp_1,p_2,p_3)=p_1p_2p_3(1+x(1-p_1)(1-p_2)(1-p_3))\quad\text{for }x \in[-1,1].\]
Then the domination functions of
$\tau_1(\textbf{\textit{X}})=\min\{X_1,X_2,X_3\}$
and
$\tau_2(\textbf{\textit{Y}})=\min\{Y_1,\max\{Y_2,Y_3\}\}$
are given by
\begin{align*}
h_1(\kern1ptp)=C(\kern1ptp,p,p)=p^3(1+x(1-p)^3),
\end{align*}
and
\[h_2(\kern1ptp)=2C(\kern1ptp,p,1)-C(\kern1ptp,p,p)=2p^2-p^3-xp^3(1-p)^3,\]
respectively. Further, it could be verified that, for all
$x\in [-1,1],$
\[\dfrac{h_1(\kern1ptp)}{h_2(\kern1ptp)}=\dfrac{p^3(1+x(1-p)^3)}{2p^2-p^3(1+x(1-p)^3)}\text{ is increasing in $p\in(0,1)$,}\]
and
\[p\dfrac{h_2'(\kern1ptp)}{h_2(\kern1ptp)}=\dfrac{4-3(1+x)p+12xp^2-15xp^3+6xp^4}{2-(1+x)p+3xp^2-3xp^3+xp^4} \text{ is decreasing in $p\in(0,1)$,}\]
and hence our claim is proved.
In the next theorem, we prove the same result as in Theorem 4.5, but for the reversed hazard rate order. The proof follows from [1, Corollary 3.2] and Theorem 3.6.
Theorem 4.6. Let
$\textbf{\textit{X}}=(X_1,X_2,\ldots,X_{n})$
and
$\textbf{\textit{Y}}=(Y_1,Y_2,\ldots,Y_{m})$
be two sets of components’ lifetimes. Assume the
$X_i$
are identical and the
$Y_j$
are identical. Suppose that
$\{\mathrm{(i)},\mathrm{(ii)},\mathrm{(iii)},\mathrm{(v)}\}$
or
$\{\mathrm{(i)},\mathrm{(ii)},\mathrm{(iv)},\mathrm{(v)}\}$
holds:
(i)
$(1-h_1(\kern1ptp))/(1- h_2(\kern1ptp))$
is increasing in
$p\in (0,1)$
,(ii)
$(1-p)h_1'(\kern1ptp)/(1-h_1(\kern1ptp))$
is increasing in
$p\in (0,1)$
,(iii)
$X_1(\theta_1)\leq_{\text{rhr}}X_1(\theta_2)\leq_{\text{rhr}}Y_1(\theta_2)\leq_{\text{rhr}}Y_1(\theta_1)$
for all
$\theta_1, \theta_2\in \Omega\text{ such that }\theta_1\leq \theta_2$
,(iv)
$X_1(\theta_2)\leq_{\text{rhr}}X_1(\theta_1)\leq_{\text{rhr}}Y_1(\theta_1)\leq_{\text{rhr}}Y_1(\theta_2)$
for all
$\theta_1, \theta_2\in \Omega\text{ such that }\theta_1\leq \theta_2$
,(v)
$\Theta_1\leq_{\text{rhr}} \Theta_2$
.
Then
$\tau_1 (\textbf{\textit{X}}(\Theta_1)) \leq_{\text{rhr}}\tau_2 (\textbf{\textit{Y}}(\Theta_2)) $
.
5. One of the coherent systems in a random environment
In this section as previously, we compare two coherent systems with respect to different stochastic orders. However, we assume that one of them operates in a random environment, whereas the other operates in a deterministic environment. As already mentioned, although this case can be viewed as the special case of the discussion in the previous section, technically it is more convenient to consider it independently. Let
$\tau_1 (\textbf{\textit{X}}(\Theta)) $
be the lifetime of a coherent system that operates in a random environment modelled by a random variable
$\Theta$
with support
$\Omega$
. Further, let
$\tau_2 (\textbf{\textit{Y}}) $
be the lifetime of the other coherent system that operates in some baseline, deterministic environment. For a given environment
$\Theta=\theta$
, we denote the domination function of
$\tau_1 (\textbf{\textit{X}}(\theta)) $
by
$h_1(\!\cdot\!)$
. Further, we denote the domination function of
$\tau_2 (\textbf{\textit{Y}}) $
by
$h_2(\!\cdot\!)$
.
5.1. Systems with not necessarily identical components
In the following theorem we show that under a set of sufficient conditions
$\tau_1 (\textbf{\textit{X}}(\Theta)) $
dominates
$\tau_2 (\textbf{\textit{Y}}) $
with respect to the usual stochastic order. The proof follows along the same lines as in Theorem 3.1 and is hence omitted.
Theorem 5.1. Let
$\textbf{\textit{X}}=(X_1,X_2,\ldots,X_{n})$
and
$\textbf{\textit{Y}}=(Y_1,Y_2,\ldots,Y_{m})$
be two sets of components’ lifetimes. Suppose that the following conditions hold:
(i)
$h_1(\kern1ptp_1,p_2,\ldots,p_n)\leq h_2(\kern1ptp_1,p_2,\ldots,p_m)$
,(ii)
$X_i(\theta)\leq_{\text{st}}Y_i $
for all
$\theta\in \Omega,$
and for all
$i=1,2,\ldots,\min\{m,n\}.$
Then
$\tau_1 (\textbf{\textit{X}}(\Theta)) \leq_{\text{st}}\tau_2 (\textbf{\textit{Y}}) $
.
The following theorem provides some sufficient conditions for proving that
$\tau_1 (\textbf{\textit{X}}(\Theta)) $
is smaller than
$\tau_2 (\textbf{\textit{Y}}) $
with respect to the hazard rate order. See the Appendix for the proof.
Theorem 5.2. Let
$\textbf{\textit{X}}=(X_1,X_2,\ldots,X_{n})$
and
$\textbf{\textit{Y}}=(Y_1,Y_2,\ldots,Y_{m})$
be two sets of components’ lifetimes. Suppose that
$\{\mathrm{(i)},\mathrm{(ii)},\mathrm{(iv)}\}$
or
$\{\mathrm{(i)},\mathrm{(iii)},\mathrm{(iv)}\}$
holds:
(i)
$\displaystyle\sum_{i=1}^n\dfrac{p_i}{h_1 (\textbf{\textit{\:p}}) }\dfrac{\partial h_1 (\textbf{\textit{\:p}}) }{\partial p_i}\geq \sum_{i=1}^m\dfrac{p_i}{{h_2} (\textbf{\textit{\:p}}) }\dfrac{\partial {h_2} (\textbf{\textit{\:p}}) }{\partial p_i}$
,(ii)
$\displaystyle\sum_{i=1}^n\dfrac{p_i}{h_1 (\textbf{\textit{\:p}}) }\dfrac{\partial h_1 (\textbf{\textit{\:p}}) }{\partial p_i}$
is decreasing in each
$p_i$
,
$i=1,2,\ldots,n$
,(iii)
$\displaystyle\sum_{i=1}^m\dfrac{p_i}{h_2 (\textbf{\textit{\:p}}) }\dfrac{\partial h_2 (\textbf{\textit{\:p}}) }{\partial p_i}$
is decreasing in each
$p_i$
,
$i=1,2,\ldots,m$
,(iv)
$X_i(\theta)\leq_{\text{hr}}Y_j $
for all
$\theta\in \Omega,$
and for all
$i=1,2,\ldots,n$
and
$j=1,2,\ldots,m.$
Then
$\tau_1 (\textbf{\textit{X}}(\Theta)) \leq_{\text{hr}}\tau_2 (\textbf{\textit{Y}}) $
.
Now we discuss the corresponding result for the reversed hazard rate order. The proof follows along the same lines as in Theorem 5.2 and is hence omitted.
Theorem 5.3. Let
$\textbf{\textit{X}}=(X_1,X_2,\ldots,X_{n})$
and
$\textbf{\textit{Y}}=(Y_1,Y_2,\ldots,Y_{m})$
be two sets of components’ lifetimes. Suppose that
$\{\mathrm{(i)},\mathrm{(ii)},\mathrm{(iv)}\}$
or
$\{\mathrm{(i)},\mathrm{(iii)},\mathrm{(iv)}\}$
holds:
(i)
$\displaystyle\sum_{i=1}^n\dfrac{1-p_i}{1-h_1 (\textbf{\textit{\:p}}) }\dfrac{\partial h_1 (\textbf{\textit{\:p}}) }{\partial p_i}\leq \sum_{i=1}^m\dfrac{1-p_i}{1-{h_2} (\textbf{\textit{\:p}}) }\dfrac{\partial {h_2} (\textbf{\textit{\:p}}) }{\partial p_i}$
,(ii)
$\displaystyle\sum_{i=1}^n\dfrac{1-p_i}{1-h_1 (\textbf{\textit{\:p}}) }\dfrac{\partial h_1 (\textbf{\textit{\:p}}) }{\partial p_i}$
is increasing in each
$p_i$
,
$i=1,2,\ldots,n$
,(iii)
$\displaystyle\sum_{i=1}^m\dfrac{1-p_i}{1-h_2 (\textbf{\textit{\:p}}) }\dfrac{\partial h_2 (\textbf{\textit{\:p}}) }{\partial p_i}$
is increasing in each
$p_i$
,
$i=1,2,\ldots,m$
,(iv)
$X_i(\theta)\leq_{\text{rhr}}Y_j $
for all
$\theta\in \Omega,$
and for all
$i=1,2,\ldots,n$
and
$j=1,2,\ldots,m.$
Then
$\tau_1 (\textbf{\textit{X}}(\Theta)) \leq_{\text{rhr}}\tau_2 (\textbf{\textit{Y}}) $
.
Below we discuss the corresponding result for the likelihood ratio order. See the Appendix for the proof.
Theorem 5.4. Let
$\textbf{\textit{X}}=(X_1,X_2,\ldots,X_{n})$
and
$\textbf{\textit{Y}}=(Y_1,Y_2,\ldots,Y_{m})$
be two sets of components’ lifetimes. Suppose that the following conditions hold:
(i) for all
$i=1,2,\ldots,n \text{ and }j=1,2,\ldots,m,$
where
$$
{\frac{\partial {{h}_{2}}(q)}{\partial {{q}_{j}}}}/{\frac{\partial {{h}_{1}}(\;p)}{\partial {{p}_{i}}}}\;\text{is}\,\text{increasing}\,\text{in}\,x\,\text{for}\,\text{all}\,\theta \in \Omega ,
$$
$p_i=\skew5\bar{F}_{X_i}(x\!{\mid}\!\theta)$
,
$q_j=\skew5\bar{F}_{Y_j}(x)$
,(ii)
$X_i(\theta)\leq_{\text{lr}}Y_j $
for all
$\theta\in \Omega,$
and for all
$i=1,2,\ldots,n$
and
$j=1,2,\ldots,m.$
Then
$\tau_1 (\textbf{\textit{X}}(\Theta)) \leq_{\text{lr}}\tau_2 (\textbf{\textit{Y}}) $
.
5.2. Systems with identical components
In this subsection we consider coherent systems of identical components. For brevity, we only discuss the corresponding result for the likelihood ratio order. Comparisons with other stochastic orders can easily be derived similar to the previous section. In the following theorems, we compare
$\tau_1 (\textbf{\textit{X}}(\Theta)) $
and
$\tau_2 (\textbf{\textit{Y}}) $
with respect to the likelihood ratio order. The proof could be completed along the same lines as in Theorem 5.4 and is hence omitted.
Theorem 5.5. Let
$\textbf{\textit{X}}=(X_1,X_2,\ldots,X_{n})$
and
$\textbf{\textit{Y}}=(Y_1,Y_2,\ldots,Y_{m})$
be two sets of components’ lifetimes. Assume the
$X_i$
are identical and the
$Y_j$
are identical. Suppose that the following conditions hold:
(i)
$h_1'(\kern1ptp)/h_2'(\kern1ptp)$
is increasing in
$p\in (0,1)$
,(ii) for
$k=1$
or 2, there exists some point
$\mu \in (0,1)$
such that(a)
$ph''_k(\kern1ptp)/h'_k(\kern1ptp)$
is decreasing and positive for all
$p \in(0,\mu),$
and(b)
$(1-p)h''_k(\kern1ptp)/h'_k(\kern1ptp)$
is decreasing and negative for all
$ p \in(\mu,1)$
,
(iii)
$X_1(\theta)\leq_{\text{lr}}Y_1 $
for all
$\theta\in \Omega$
.
Then
$\tau_1 (\textbf{\textit{X}}(\Theta)) \leq_{\text{lr}}\tau_2 (\textbf{\textit{Y}}) $
.
6. Concluding remarks
In this paper we study the impact of a random environment on lifetimes of coherent systems with dependent lifetimes.
Motivated by discussions in [Reference Amini-Seresht, Zhang and Balakrishnan1], we present solutions for some open problems formulated in this paper. We also generalize the results of these authors and present some new comparisons as well. Specifically, we provide different sets of sufficient conditions for one system to dominate the other with respect to different stochastic orders, namely, usual stochastic order, hazard rate order, reversed hazard rate order, and likelihood ratio order.
Even though we have incorporated a large number of new results in this paper, there are still some open problems left over. One of them is to generalize the results discussed in Sections 3 and 4 to the likelihood ratio order.
We conclude our discussion by mentioning the fact that the straightforward corollaries corresponding to Theorems 4.1, 4.4, 4.5, 4.6, 5.1, 5.2, 5.3, and 5.4 could be formulated (by using Lemmas 2.2–2.6) similarly to the corollary given below for Theorem 4.1.
Corollary 6.1. Let
$\textbf{\textit{X}}=(X_1,X_2,\ldots,X_{n})$
and
$\textbf{\textit{Y}}=(Y_1,Y_2,\ldots,Y_{m})$
be two sets of components’ lifetimes. Assume the
$X_i$
are independent and the
$Y_j$
are independent. Suppose that the set of conditions
$\{\mathrm{(ii)},\mathrm{(iv)}\}$
or
$\{\mathrm{(iii)},\mathrm{(iv)}\}$
in Theorem 4.1 holds:
(i)
${\tau_{k:n} (\textbf{\textit{X}}(\Theta_1)) }\leq_{\text{st}}$
${\tau_{l:n} (\textbf{\textit{Y}}(\Theta_2)) }$
for
$l\leq k$
,(ii)
${\tau_{k:n} (\textbf{\textit{X}}(\Theta_1)) }\leq_{\text{st}}$
${\tau_{k:m} (\textbf{\textit{Y}}(\Theta_2)) }$
for
$n\leq m$
,(iii)
${\tau_{k:n} (\textbf{\textit{X}}(\Theta_1)) }\leq_{\text{st}}$
${\tau_{k-r:n-r} (\textbf{\textit{Y}}(\Theta_2)) }$
for
$r\leq k$
,(iv)
${\tau_{k:n} (\textbf{\textit{X}}(\Theta_1)) }\leq_{\text{st}}$
${\tau_{l:m} (\textbf{\textit{Y}}(\Theta_2)) }$
for
$l\leq k$
and
$n-k\leq m-l$
.
Appendix A
Proof of Theorem 3.1. Let
$k=\min\{m,n\}$
. Note that
\begin{align*}
\skew5\bar{F}_{\tau_1 (\textbf{\textit{X}}(\Theta)) }(x)
&= \int_\Omega h_1 (\skew5\bar{F}_{X_1}(x\!{\mid}\!\theta),\skew5\bar{F}_{X_2}(x\!{\mid}\!\theta),\ldots,\skew5\bar{F}_{X_n}(x\!{\mid}\!\theta)) \,{\textrm{d}} F _{\Theta}(\theta)
\\*
&\leq \int_\Omega h_1 (\skew5\bar{F}_{Y_1}(x\!{\mid}\!\theta),\skew5\bar{F}_{Y_2}(x\!{\mid}\!\theta),\ldots,\skew5\bar{F}_{Y_k}(x\!{\mid}\!\theta)) \,{\textrm{d}} F _{\Theta}(\theta)
\\
&\leq \int_\Omega h_2 (\skew5\bar{F}_{Y_1}(x\!{\mid}\!\theta),\skew5\bar{F}_{Y_2}(x\!{\mid}\!\theta),\ldots,\skew5\bar{F}_{Y_m}(x\!{\mid}\!\theta)) \,{\textrm{d}} F _{\Theta}(\theta)
\\*
&= \skew5\bar{F}_{\tau_2 (\textbf{\textit{Y}}(\Theta)) }(x),
\end{align*}
where the first inequality follows from condition (ii) and the fact that
$h_1 (\textbf{\textit{\:p}}) $
is increasing in each
$p_i$
. The second inequality follows from condition (i). Hence the result is proved.
Proof of Theorem 3.2. We only prove the result under the set of conditions
$\{\mathrm{(i)},\mathrm{(ii)},\mathrm{(iii)}\}$
. The result could be proved along the same lines under the second set of conditions. Note that
\begin{align*}
\sum_{i=1}^m r_{Y_i}(x\!{\mid}\!\theta_2)\bigg[\dfrac{p_i}{h_2 (\textbf{\textit{\:p}}) }\dfrac{\partial h_2 (\textbf{\textit{\:p}}) }{\partial p_i}\bigg]_{p_i=\skew5\bar{F}_{Y_i}(x\!{\mid}\!\theta_2)}
\geq \sum_{i=1}^m r_{Y_i}(x\!{\mid}\!\theta_1)\bigg[\dfrac{q_i}{h_2 (\textbf{\textit{q}}) }\dfrac{\partial h_2 (\textbf{\textit{q}}) }{\partial q_i}\bigg]_{q_i=\skew5\bar{F}_{Y_i}(x\!{\mid}\!\theta_1)},
\end{align*}
which follows from [Reference Navarro, Águila, Sordo and Suárez-Liorens26, Proposition 2.3(ii)] by using conditions (ii) and (iii). The above inequality is equivalent to the fact that
\begin{align*}
\dfrac{h_2 (\skew5\bar{F}_{Y_1}(x\!{\mid}\!\theta_2),\skew5\bar{F}_{Y_2}(x\!{\mid}\!\theta_2),\ldots,\skew5\bar{F}_{Y_m}(x\!{\mid}\!\theta_2)) }{h_2 (\skew5\bar{F}_{Y_1}(x\!{\mid}\!\theta_1),\skew5\bar{F}_{Y_2}(x\!{\mid}\!\theta_1),\ldots,\skew5\bar{F}_{Y_m}(x\!{\mid}\!\theta_1)) }\text{ is decreasing in $x>0$,}
\end{align*}
or equivalently
\begin{align}
h_2 (\skew5\bar{F}_{Y_1}(x\!{\mid}\!\theta),\skew5\bar{F}_{Y_2}(x\!{\mid}\!\theta),\ldots,\skew5\bar{F}_{Y_m}(x\!{\mid}\!\theta)) \text{ is $\text{RR}_2$ in $(x,\theta)\in(0,\infty)\times \Omega$.}
\label{eqn7}
\end{align}
Further, we have
\begin{align*}
\sum_{i=1}^n r_{X_i}(x\!{\mid}\!\theta)\bigg[\dfrac{p_i}{h_1 (\textbf{\textit{\:p}}) }\dfrac{\partial h_1 (\textbf{\textit{\:p}}) }{\partial p_i}\bigg]_{p_i=\skew5\bar{F}_{X_i}(x\!{\mid}\!\theta)}
&\geq \sum_{i=1}^m r_{X_i}(x\!{\mid}\!\theta)\bigg[\dfrac{p_i}{h_1 (\textbf{\textit{\:p}}) }\dfrac{\partial h_1 (\textbf{\textit{\:p}}) }{\partial p_i}\bigg]_{p_i=\skew5\bar{F}_{X_i}(x\!{\mid}\!\theta)}
\\*
&\geq \sum_{i=1}^m r_{Y_i}(x\!{\mid}\!\theta)\bigg[\dfrac{p_i}{h_2 (\textbf{\textit{\:p}}) }\dfrac{\partial h_2 (\textbf{\textit{\:p}}) }{\partial p_i}\bigg]_{p_i=\skew5\bar{F}_{X_i}(x\!{\mid}\!\theta)}
\\*
&\geq \sum_{i=1}^m r_{Y_i}(x\!{\mid}\!\theta)\bigg[\dfrac{q_i}{h_2 (\textbf{\textit{q}}) }\dfrac{\partial h_2 (\textbf{\textit{q}}) }{\partial q_i}\bigg]_{q_i=\skew5\bar{F}_{Y_i}(x\!{\mid}\!\theta)},
\end{align*}
where the first inequality holds because each term in the summation is non-negative. The second inequality follows from conditions (i) and (iii), whereas the third inequality follows from (ii) and (iii). Thus the above expression can equivalently be written as
\begin{align}
\dfrac{h_2 (\skew5\bar{F}_{Y_1}(x\!{\mid}\!\theta),\skew5\bar{F}_{Y_2}(x\!{\mid}\!\theta),\ldots,\skew5\bar{F}_{Y_m}(x\!{\mid}\!\theta)) }{h_1 (\skew5\bar{F}_{X_1}(x\!{\mid}\!\theta),\skew5\bar{F}_{X_2}(x\!{\mid}\!\theta),\ldots,\skew5\bar{F}_{X_n}(x\!{\mid}\!\theta)) }\text{ is increasing in $x>0 $ for all $\theta \in \Omega$.}
\label{eqn8}
\end{align}
Again, condition (iii) implies that, for all
$ \theta_1\leq \theta_2$
and for all
$i=1,2,\ldots,n$
and
$j=1,2,\ldots,m,$
$$
X_i(\theta_1)\leq_{\text{st}}X_i(\theta_2)\quad\text{and}\quad Y_j(\theta_2)\leq_{\text{st}}Y_j(\theta_1).
$$}
Using this, we obtain
\[
\dfrac{\textrm{d}}{\textrm{d} \theta }{h_1 (\skew5\bar{F}_{X_1}(x\!{\mid}\!\theta),\skew5\bar{F}_{X_2}(x\!{\mid}\!\theta),\ldots,\skew5\bar{F}_{X_n}(x\!{\mid}\!\theta)) } = \sum_{i=1}^n\bigg[\dfrac{\partial h_1 (\textbf{\textit{\:p}}) }{\partial p_i}\dfrac{\textrm{d} p_i}{\textrm{d} \theta }\bigg]_{p_i=\skew5\bar{F}_{X_i}(x\!{\mid}\!\theta)}
\geq 0
\]
and
\[
\dfrac{{\textrm{d}}}{{\textrm{d}} \theta }{h_2 (\skew5\bar{F}_{Y_1}(x\!{\mid}\!\theta),\skew5\bar{F}_{Y_2}(x\!{\mid}\!\theta),\ldots,\skew5\bar{F}_{Y_m}(x\!{\mid}\!\theta)) } = \sum_{j=1}^m\bigg[\dfrac{\partial h_2 (\textbf{\textit{\:p}}) }{\partial p_i}\dfrac{{\textrm{d}} p_i}{{\textrm{d}} \theta }\bigg]_{p_i=\skew5\bar{F}_{Y_i}(x\!{\mid}\!\theta)}
\leq 0,
\]
or equivalently
\begin{align*}
\dfrac{1}{h_1 (\skew5\bar{F}_{X_1}(x\!{\mid}\!\theta),\skew5\bar{F}_{X_2}(x\!{\mid}\!\theta),\ldots,\skew5\bar{F}_{X_n}(x\!{\mid}\!\theta)) }\text{ is decreasing in $\theta \in \Omega$}
\end{align*}
and
\begin{align*}
{h_2 (\skew5\bar{F}_{Y_1}(x\!{\mid}\!\theta),\skew5\bar{F}_{Y_2}(x\!{\mid}\!\theta),\ldots,\skew5\bar{F}_{Y_m}(x\!{\mid}\!\theta)) }\text{ is decreasing in $\theta \in \Omega$.}
\end{align*}
Combining these two, we obtain
\begin{align}
\dfrac{h_2 (\skew5\bar{F}_{Y_1}(x\!{\mid}\!\theta),\skew5\bar{F}_{Y_2}(x\!{\mid}\!\theta),\ldots,\skew5\bar{F}_{Y_m}(x\!{\mid}\!\theta)) }{h_1 (\skew5\bar{F}_{X_1}(x\!{\mid}\!\theta),\skew5\bar{F}_{X_2}(x\!{\mid}\!\theta),\ldots,\skew5\bar{F}_{X_n}(x\!{\mid}\!\theta)) }\text{ is decreasing in $\theta \in \Omega$ for all $ x>0$.}
\label{eqn9}
\end{align}
Using (A.1), (A.2), and (A.3) in Lemma 2.1, we obtain, for
$x_1\leq x_2$
,
\begin{align*}
& \dfrac{\int_\Omega h_1 (\skew5\bar{F}_{X_1}(x_2|\theta),\skew5\bar{F}_{X_2}(x_2|\theta),\ldots,\skew5\bar{F}_{X_n}(x_2|\theta)) \,{\textrm{d}} F _{\Theta}(\theta)}{\int_\Omega h_2 (\skew5\bar{F}_{Y_1}(x_2|\theta),\skew5\bar{F}_{Y_2}(x_2|\theta),\ldots,\skew5\bar{F}_{Y_m}(x_2|\theta)) \,{\textrm{d}} F _{\Theta}(\theta)}
\\*
& \qquad \leq \dfrac{\int_\Omega h_1 (\skew5\bar{F}_{X_1}(x_1|\theta),\skew5\bar{F}_{X_2}(x_1|\theta),\ldots,\skew5\bar{F}_{X_n}(x_2|\theta)) \,{\textrm{d}} F _{\Theta}(\theta)}{\int_\Omega h_2 (\skew5\bar{F}_{Y_1}(x_1|\theta),\skew5\bar{F}_{Y_2}(x_1|\theta),\ldots,\skew5\bar{F}_{Y_m}(x_1|\theta)) \,{\textrm{d}} F _{\Theta}(\theta)},
\end{align*}
or equivalently
\begin{align*}
\dfrac{\skew5\bar{F}_{\tau_1 (\textbf{\textit{X}}(\Theta)) }(x)}{\skew5\bar{F}_{\tau_2 (\textbf{\textit{Y}}(\Theta)) }(x)}=\dfrac{\int_\Omega h_1 (\skew5\bar{F}_{X_1}(x\!{\mid}\!\theta),\skew5\bar{F}_{X_2}(x\!{\mid}\!\theta),\ldots,\skew5\bar{F}_{X_n}(x\!{\mid}\!\theta)) \,{\textrm{d}} F _{\Theta}(\theta)}{\int_\Omega h_2 (\skew5\bar{F}_{Y_1}(x\!{\mid}\!\theta),\skew5\bar{F}_{Y_2}(x\!{\mid}\!\theta),\ldots,\skew5\bar{F}_{Y_m}(x\!{\mid}\!\theta)) \,{\textrm{d}} F _{\Theta}(\theta)}\text{ is decreasing in $x>0$,}
\end{align*}
and hence
$\tau_1 (\textbf{\textit{X}}(\Theta)) \leq_{\text{hr}}\tau_2 (\textbf{\textit{Y}}(\Theta)) $
.
Proof of Theorem 5.2. We only prove the result under the condition
$\{\mathrm{(i)},\mathrm{(iii)},\mathrm{(iv)}\}$
. The result follows similarly for the other case. Now, from condition (iv), we have
\begin{align}
\min_{1\leq i\leq n}r_{X_i}(x\!{\mid}\!\theta)\geq \max_{1\leq i\leq m}r_{Y_i}(x)\quad\text{for all $\theta \in \Omega$.}
\label{eqn10}
\end{align}
Then
$$\eqalign{&\sum_{i=1}^n r_{X_i}(x\!{\mid}\!\theta)\bigg[\dfrac{p_i}{h_1 (\textbf{\textit{\:p}}) }\dfrac{\partial h_1 (\textbf{\textit{\:p}}) }{\partial p_i}\bigg]_{p_i=\skew5\bar{F}_{X_i}(x\!{\mid}\!\theta)} \cr &\qquad \geq \min_{1\leq i\leq n}r_{X_i}(x\!{\mid}\!\theta)\sum_{i=1}^n \bigg[\dfrac{p_i}{h_1 (\textbf{\textit{\:p}}) }\dfrac{\partial h_1 (\textbf{\textit{\:p}}) }{\partial p_i}\bigg]_{p_i=\skew5\bar{F}_{X_i}(x\!{\mid}\!\theta)} \cr & \qquad \geq \max_{1\leq i\leq m}r_{Y_i}(x)\sum_{i=1}^m \bigg[\dfrac{p_i}{h_2 (\textbf{\textit{\:p}}) }\dfrac{\partial h_2 (\textbf{\textit{\:p}}) }{\partial p_i}\bigg]_{\substack{p_i=\skew5\bar{F}_{X_i}(x\!{\mid}\!\theta),\;i=1,2,\ldots,\min\{m,n\}\\ \{p_i=\skew5\bar{F}_{Y_i}(x),\;i=n+1,\ldots,m\}I_{[m\gt{n}]}}} \cr &\qquad \geq \max_{1\leq i\leq m}r_{Y_i}(x)\sum_{i=1}^m\nonumber \bigg[\dfrac{q_i}{h_2 (\textbf{\textit{q}}) }\dfrac{\partial h_2 (\textbf{\textit{q}}) }{\partial q_i}\bigg]_{q_i=\skew5\bar{F}_{Y_i}(x)} \cr &\qquad \geq \sum_{i=1}^m r_{Y_i}(x)\bigg[\dfrac{q_i}{h_2 (\textbf{\textit{q}}) }\dfrac{\partial h_2 (\textbf{\textit{q}}) }{\partial q_i}\bigg]_{q_i=\skew5\bar{F}_{Y_i}(x)},}$$
where the first and the fourth inequalities are obvious. The second inequality follows from (A.4) and condition (i), whereas the third inequality follows from conditions (iii) and (iv). Now, the above inequality implies that
$$\eqalign{&h_2 (\skew5\bar{F}_{Y_1}(x),\skew5\bar{F}_{Y_2}(x),\ldots,\skew5\bar{F}_{Y_m}(x)) \int_\Omega \bigg(\sum_{i=1}^n r_{X_i}(x\!{\mid}\!\theta)\bigg[{p_i}\dfrac{\partial h_1 (\textbf{\textit{\:p}}) }{\partial p_i}\bigg]_{p_i=\skew5\bar{F}_{X_i}(x\!{\mid}\!\theta)}\bigg)\,{\textrm{d}} F _{\Theta}(\theta)\cr &\quad \geq \bigg(\sum_{i=1}^m r_{Y_i}(x)\bigg[{q_i}\dfrac{\partial h_2 (\textbf{\textit{q}}) }{\partial q_i}\bigg]_{q_i=\skew5\bar{F}_{Y_i}(x)}\bigg)\int_\Omega h_1 (\skew5\bar{F}_{X_1}(x\!{\mid}\!\theta),\skew5\bar{F}_{X_2}(x\!{\mid}\!\theta),\ldots,\skew5\bar{F}_{X_n}(x\!{\mid}\!\theta)) \,{\textrm{d}} F _{\Theta}(\theta),}$$
which is equivalent to the fact that
\begin{align*}
\dfrac{\skew5\bar{F}_{\tau_1 (\textbf{\textit{X}}(\Theta)) }(x)}{\skew5\bar{F}_{\tau_2 (\textbf{\textit{Y}}) }(x)}=\dfrac{\int_\Omega h_1 (\skew5\bar{F}_{X_1}(x\!{\mid}\!\theta),\skew5\bar{F}_{X_2}(x\!{\mid}\!\theta),\ldots,\skew5\bar{F}_{X_n}(x\!{\mid}\!\theta)) \,{\textrm{d}} F _{\Theta}(\theta)}{ h_2 (\skew5\bar{F}_{Y_1}(x),\skew5\bar{F}_{Y_2}(x),\ldots,\skew5\bar{F}_{Y_m}(x)) }\text{ is decreasing in $x>0$,}
\end{align*}
and hence
$\tau_1 (\textbf{\textit{X}}(\Theta)) \leq_{\text{hr}}\tau_2 (\textbf{\textit{Y}}) $
.
Proof of Theorem 5.4. Note that
$\tau_1 (\textbf{\textit{X}}(\Theta)) \leq_{\text{lr}}\tau_2 (\textbf{\textit{Y}}) $
holds if
$$
\frac{{{f}_{{{\tau }_{1}}(X(\Theta ))}}(x)}{{{f}_{{{\tau }_{2}}(Y)}}(x)}=\text{ }{(\int_{\Omega }{[}\sum\limits_{i=1}^{n}{(}{{f}_{{{X}_{i}}}}(x\,\,\mid \,\theta )\frac{\partial {{h}_{1}}(\;p)}{\partial {{p}_{i}}})]\text{d}{{F}_{\Theta }}(\theta ))}/{(\sum\limits_{i=1}^{m}{(}{{f}_{{{Y}_{i}}}}(x)\frac{\partial {{h}_{2}}(q)}{\partial {{q}_{i}}}))\text{ }}\;\text{is}\,\text{decreasing}\,\text{in}\,x\gt\text{0},
$$
or equivalently
$$
{(\sum\limits_{i=1}^{n}{[}\int_{\Omega }{(}{{f}_{{{X}_{i}}}}(x\mid \theta )\frac{\partial {{h}_{1}}(\;p)}{\partial {{p}_{i}}})\text{d}{{F}_{\Theta }}(\theta )])}/{(\sum\limits_{i=1}^{m}{(}{{f}_{{{Y}_{i}}}}(x)\frac{\partial {{h}_{2}}(q)}{\partial {{q}_{i}}}))}\;\,\text{is}\,\text{decreasing}\,\text{in}\,x\gt\text{0}.
$$
This holds if, for all
$i=1,2,\ldots,n$
and
$j=1,2,\ldots,m$
,
$$
\int_{\Omega }{\left( \frac{{{f}_{{{X}_{i}}}}(x\mid \theta )}{{{f}_{{{Y}_{j}}}}(x)} \right)}\left( {\frac{\partial {{h}_{1}}(\;p)}{\partial {{p}_{i}}}}/{\frac{\partial {{h}_{2}}(q)}{\partial {{q}_{j}}}}\; \right)\text{d}{{F}_{\Theta }}(\theta )\,\text{is}\,\text{decreasing}\,\text{in}\,x\gt\text{0},
$$
which holds if, for all
$i=1,2,\ldots,n$
and
$j=1,2,\ldots,m$
,
$$
\dfrac{f_{X_i}(x\!{\mid}\!\theta)}{f_{Y_j}(x)} \text{ is decreasing in $x\gt0$ for all $ \theta \in \Omega$}$$
and
$${\frac{\partial {{h}_{1}}(\;p)}{\partial {{p}_{i}}}}/{\frac{\partial {{h}_{2}}(q)}{\partial {{q}_{j}}}}\;\text{is}\,\text{decreasing in }x\gt\text{0},$$
which are true because of conditions (i) and (ii).
Acknowledgements
The authors are grateful to the Editor-in-Chief, the Associate Editor, and the anonymous reviewers for their valuable constructive comments, which led to an improved version of the manuscript. The first author sincerely acknowledges the financial support from IIT Jodhpur. The work of the second author was supported by National Research Foundation of South Africa (grant 103613).
