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Stochastic ordering results on extreme order statistics from dependent samples with Archimedean copula
Journal of Inequalities and Applications volume 2024, Article number: 120 (2024)
Abstract
This paper considers parallel and series systems with heterogeneous components having dependent exponential lifetimes. The underlying dependence is assumed to be Archimedean and the component lifetimes are supposed to be connected according to an Archimedean copula. Sufficient conditions are found to dominate a parallel system with heterogenous exponential components, with respect to the dispersive order, by another parallel system with homogenous exponential components where the dependence structure between lifetimes of components is the same. We also compare two series systems (and two parallel systems) with general oneparameter dependent components and with respect to the usual stochastic ordering. Examples are given to illustrate the theoretical findings.
1 Introduction
Numerous researchers have examined stochastic comparisons between order statistics from specific families of distributions in the literature when the underlying random variables are either independent and identically distributed (i.i.d.) or independent but not identically distributed (i.n.i.d.). The reader is directed, among other sources, to [2] for a review of this subject. Stochastic orders of order statistics when underlying random variables induce certain dependent structures were examined by some authors. In this manuscript, we additionally validate several novel stochastic order outcomes amidst the most extensive order statistics. These results extend several established findings from i.n.i.d. to scenarios in which the random variables lack both identical distribution and independence. Order statistics constitute an important class of statistics in reliability and survival analysis to model the lifetime events. They have also been used frequently in the context of many reliability or survival models for the purpose of estimation of the parameters of the models. In the context of reliability engineering, a system with a koutofn structure is a system which needs \((nk+1)\) from its n components to be operational in order for the entire system to work, where \(k=1,\ldots ,n\). The koutofn system structure is a very popular type of redundancy in faulttolerant systems. It finds wide applications in both industrial and military systems. Two cases of \(k=1\) and \(k=n\) which correspond to two extreme order statistics, i.e., \(X_{1:n}\) and \(X_{n:n}\), are wellknown as the series and parallel system, respectively. Stochastic comparisons of series and parallel systems have been an attractive topic of research during recent decades (see, for instance, [6–10, 12, 14, 18], and [15]). For some stochastic comparisons of ordered random variables in the i.i.d. case, we refer the reader to [1, 4], and [5]. We first go over the definition of a copula function and also recall some concepts of stochastic orders before getting into their specific roles in our investigations.
Let us first recall some stochastic orders. Throughout the paper, increasing means nondecreasing and decreasing means nonincreasing, respectively. All integrals and expectations are assumed to exist whenever they appear.
Definition 1.1
Let X and Y be two nonnegative random variables with density functions f and g, distribution functions F and G, survival functions \(\overline{F}=1F\) and \(\overline{G}=1G\), hazard rate functions \({r_{X}}=f/\overline{F}\) and \({r_{Y}}=g/\overline{G}\), respectively. Then, X is said to be smaller than Y in the

(i)
usual stochastic order (denoted by \(X\leq _{\mathrm{st}} Y\)) if \(\overline{F}(x)\leq \overline{G}(x)\) for all \(x\in{\mathbb{R}}_{+}\), or equivalently, \(\mathbb{E}[w(X)]\leq [\geq ] \mathbb{E}[w(Y)]\) for any increasing [decreasing] function \(w:{\mathbb{R}}\rightarrow{\mathbb{R}}\);

(ii)
hazard rate order (denoted by \(X\leq _{\mathrm{hr}} Y\)) if \(\overline{G}(x)/\overline{F}(x)\) is increasing in \(x\in{\mathbb{R}}_{+}\), or equivalently, \({r_{X}(x)}\geq {r_{Y}(x)}\) for all \(x\in{\mathbb{R}}_{+}\);

(iii)
dispersive order (denoted by \(X\leq _{\mathrm{disp}} Y\)) if
$$ F^{1}_{X}(\beta )F^{1}_{X}(\alpha )\leq F^{1}_{Y}(\beta )F^{1}_{Y}( \alpha )\quad \text{for}\ 0\leq \alpha < \beta \leq 1, $$or equivalently, if and only if
$$ f_{Y}(F^{1}_{Y}(F_{X}(x)))\leq f_{X}(x)\quad \text{for all } x\in{ \mathbb{R}}_{+}. $$(1)
It is wellknown that
but neither reversed hazard rate order nor hazard rate order implies the other. For comprehensive discussions on stochastic orders, one may refer to [16]. One may also refer to [13].
Now, we give some preliminaries of the concept of dependence through copula function. Let \((X_{1},\ldots ,X_{n})\) be the vector of components’ lifetimes, having joint distribution F, survival function \({ \overline{\boldsymbol{F}}}\), and marginal distributions \(F_{i}\), \(i=1,\ldots ,n\). A function \(\mathbb{C} :[0,1]^{m}\rightarrow \mathbb{R}^{+}\) is said to be a copula of \((X_{1},\ldots ,X_{n})\) if
If \(F_{i}\) are continuous, then the copula \(\mathbb{C}\) is unique, and it is defined as
where \(F_{i}^{1}\) denotes the inverse of the distribution function of the random variable \(X_{i},i=1,\ldots ,n\).
Likewise, the survival copula associated with a multivariate distribution function F is
Among copulas (or survival copulas), the class of Archimedean copulas is particularly interesting. Archimedean copulas are widely used in reliability theory and actuarial mathematics because of their mathematical tractability and wide range of dependencies. For a decreasing and continuous function \(\phi : [0, \infty ) \rightarrow [0,1]\) such that \(\phi (0)=1\) and \(\phi (\infty )=0\), where \(\psi =\phi ^{1}\) is the pseudoinverse, a copula is called Archimedean if it can be written as
where ϕ is usually called the generator of the Archimedean copula \(\mathbb{C}\), if \((1)^{k}\phi ^{[k]}(x)\geq 0\), for \(k = 0,\ldots , n2\) and \((1)^{n2}\phi ^{[n2]}(x)\) is decreasing and convex. In the above, \(\phi ^{[k]}(x)\) denotes the kth derivative of the function \(\phi (x)\) with respect to x.
Majorization is one of the key tools in mathematical statistics and applied probability, which is a preordering on vectors by sorting all components in nonincreasing order. The concepts of majorization of vectors and Schurconcavity (Schurconvexity) of functions are given as follows.
Definition 1.2
Let λ, \(\boldsymbol{\lambda}^{*}\) denote two ndimensional real vectors. Let \(\lambda _{(1)}\leq \lambda _{(2)}\leq \cdots \leq \lambda _{(n)}\), \(\lambda ^{*}_{(1)}\leq \lambda ^{*}_{(2)}\leq \cdots \leq \lambda ^{*}_{(n)}\) be their ordered components. Then

λ is said to be majorized by \(\boldsymbol{\lambda} ^{*}\), in symbols \(\boldsymbol{\lambda} \overset{m}{\preceq}\boldsymbol{\lambda} ^{*}\), if \(\sum _{i=1}^{k} \lambda ^{*}_{(i)}\leq \sum _{i=1}^{k}\lambda _{(i)}\); for \(k=1,2,\ldots ,n1\), and \(\sum _{i=1}^{n} \lambda _{(i)}=\sum _{i=1}^{n}\lambda ^{*}_{(i)}\);

λ is said to be weakly submajorized by \(\boldsymbol{\lambda} ^{*}\), in symbols \(\boldsymbol{\lambda} \preceq _{w}\boldsymbol{\lambda} ^{*}\), if \(\sum _{i=1}^{k} \lambda _{(i)}\leq \sum _{i=1}^{k}\lambda ^{*}_{(i)}\); for \(k=1,2,\ldots , n\);

λ is said to be plarger than another vector \(\boldsymbol{\lambda} ^{*}\), in symbols \(\boldsymbol{\lambda} \overset{p}{\preceq}\boldsymbol{\lambda} ^{*}\), if \(\prod _{i=1}^{j} \lambda ^{*}_{(i)}\leq \prod _{i=1}^{j} \lambda _{(i)}\), \(j=1,\ldots ,n\).
The following lemmas show that the notion of majorization is quite useful in establishing various inequalities.
Lemma 1.3
Let \(\Phi :\mathbb{R}^{n}\rightarrow \mathbb{R}\) be continuously differentiable. Necessary and sufficient conditions for Φ to be Schurconvex (concave) on \(\mathbb{R}^{n}\) are: Φ is symmetric on \(\mathbb{R}^{n}\) and for all \(i\neq j\),
Lemma 1.4
A realvalued function \(\Phi :\mathbb{R}^{n}\rightarrow \mathbb{R}\) is said to be a Schurconvex (Schurconcave) function if \(\boldsymbol{\lambda} \overset{m}{\preceq}\boldsymbol{\lambda} ^{*}\) implies \(\Phi (\boldsymbol{\lambda} )\leq (\geq )\Phi (\boldsymbol{\lambda} ^{*})\), for all \(\boldsymbol{\lambda} ,\boldsymbol{\lambda} ^{*}\in \mathbb{R}^{n}\).
Let \(X_{1},\ldots ,X_{n}\) be independent exponential random variables with \(X_{i}\), \(i=1,\ldots ,n\) having hazard rates \(\lambda _{i} > 0\) for \(i=1,\ldots ,n\). Let \(Y_{1},\ldots ,Y_{n}\) be another set of independent exponential random variables with common hazard rate \(\overline{\lambda}=\frac{1}{n}\sum _{i=1}^{n} \lambda _{i}^{*}\). [3] proved that \(X_{n:n}\geq _{\mathrm{disp}} Y_{n:n}\).
In Sect. 2, we generalize the above result to the cases where the random variables \(X_{i} ,i=1,\ldots ,n\) and the random variables \(Y_{i},i=1,\ldots ,n\) are dependent and have a common Archimedean copula with a generator function ϕ satisfying a number of mild conditions. In particular, we show that when the generator function is \(\phi (x)=\exp (x)\), the results are consistent with the case where the underlying random variables are independent. In Sect. 3, we compare two parallel systems with Archimedean copulabased dependent components with a oneparameter family of the life distribution. In Sect. 4, we continue the topic discussed in Sect. 3 for series systems. In Sect. 5, we conclude the paper by describing a number of examples, some remarks, and explanations.
2 Dispersive order of parallel systems
We bring a technical lemma which will be used in the sequel.
Lemma 2.1
Let \((1\phi (x))^{\frac{1}{d(x)}}\) be increasing in \(x\geq 0\), where \(d(\cdot )\) is a nonnegative function. Then, \(\frac{\lambda}{d(Z(t,\lambda ))}\) is increasing in \(\lambda >0\), for all \(t\geq 0\), where \(Z(t,\lambda )=\psi (1e^{\lambda t})\).
Proof
Let us write
Now, since \(\psi (1e^{\lambda t})\) is decreasing in λ, for all \(t\geq 0\), thus \(\frac{\lambda}{d(Z(t,\lambda ))}\) is increasing in λ, for all \(t\geq 0\), if
i.e., when
The above statement is also equivalent to \((1\phi (x))^{\frac{1}{d(x)}}\) being increasing in \(x \geq 0\). The proof of lemma is complete. □
Next result compares the largest order statistics of heterogeneous dependent exponential random variables with that of homogeneous dependent exponential random variables with respect to dispersive ordering.
Theorem 2.2
Let \(\boldsymbol{X}=(X_{1},\ldots ,X_{n})\) be a dependent random vector with the Archimedean copula having generator ϕ where \(X_{i}\) has exponential distribution with hazard rate \(\lambda _{i}\) and, further, let \(\boldsymbol{Y}=(Y_{1},\ldots ,Y_{n})\) be another dependent random vector having the same Archimedean copula as X where \(Y_{i}\) has exponential distribution with hazard rate λ. If there exists a nonnegative function \(d(\cdot )\) satisfying

(i)
\((1\phi (u))^{\frac{1}{d(u)}}\) is increasing (resp. decreasing) in \(u\geq 0\);

(ii)
\(d(u)\frac{1\phi (u)}{\phi '(u)}\) is decreasing (resp. increasing) and also convex in \(u\geq 0\),
then
where \(Z(x,\lambda _{i})= \psi (1e^{\lambda _{i} x})\), \(i=1,\ldots ,n\) and \(\bar{Z}(x,\boldsymbol{\lambda} )=\frac{1}{n}\sum _{i=1}^{n} Z(x,\lambda _{i})\) and \(x'=\arg \max _{x\geq 0} m(x,\boldsymbol{\lambda} )\) such that \(\lambda >0\), then
Proof
Due to Definition 1.1\((i)\), it is sufficient to prove that
The density functions of \(Y_{n:n}\) and \(X_{n:n}\), respectively, can be represented as
and
It is easy to calculate that for \(x\in \mathbb{R}^{+}\),
Thus, the lefthand side of (2) can be written as
From the assumption (ii), \(d(u)\frac{1\phi (u)}{\phi '(u)}\) is a convex function, therefore, for all \(x\in \mathbb{R}^{+}\),
Since from assumption (ii) \((1\phi (u))^{d(u)}\) is increasing (resp. decreasing) in \(u\geq 0\) and from assumption (ii) \(d(u)\frac{1\phi (u)}{\phi '(u)}\) is decreasing (resp. increasing) and, further, since \(Z(x,\lambda )\) is decreasing in λ for every \(x\in \mathbb{R}^{+}\), \(d(Z(x,\lambda )) \frac{1\phi \left (Z(x,\lambda )\right )}{\phi '\left (Z(x,\lambda )\right )}\) is increasing (resp. decreasing) with respect to λ. In addition, in view of the condition (i) and by using Lemma 2.1, \(\frac{\lambda}{d(Z(x,\lambda ))}\) is increasing in λ, for all \(x\geq 0\). By Chebyshev’s sum inequality (see, e.g., [17]), it then follows that
where the second inequality follows from equation (3) and the third from the assumption \(m(x',\boldsymbol{\lambda} ) \leq \lambda \), and the fact that \(\phi '\) is a negative function. As \(\phi '\) is a nonpositive function, it thus holds that
where the inequality follows from (4). Hence, the required result follows from (1). □
We want to remark that there is always a proper function d in Theorem 2.2 for which both conditions (i) and (ii) in this theorem are fulfilled. Specifically, if \(d(x)=\phi '(x)\), which is a nonnegative function, it is seen that \(\frac{1\phi (x)}{\phi '(x)}d(x)=\phi (x)1\), which is a decreasing convex function in \(x \geq 0\) and, in addition, \((1\phi (x))^{\frac{1}{d(x)}}\) is increasing in \(x\geq 0\). Below we give an example to indicate that the result of Theorem 2.2 is obtained as a conclusion of the conditions presented. This example shows that the result of Theorem 2.2 is applicable when there is no dependency between random variables.
Example 2.3
Let us suppose that \(\boldsymbol{X}=(X_{1},\ldots ,X_{n})\) follows the Archimedean copula with generator \(\phi (x)=\exp (x)\) where \(X_{i}\sim E(\lambda _{i})\) and let \(\boldsymbol{Y}=(Y_{1},\ldots ,Y_{n})\) follow the Archimedean copula with the same generator in which \(Y_{i}\sim E(\lambda )\). We choose \(d(x)=\phi '(x)\). Therefore, the conditions (i) and (ii) in Theorem 2.2 hold. In this particular case of the generator function, the random variables \(X_{1},\ldots ,X_{n}\) are independent and the random variables \(Y_{1},\ldots ,Y_{n}\) are also independent. In addition, we observe that
Therefore, if \(\frac{\sum _{i=1}^{n} \lambda _{i}}{n}\leq \lambda \) then, using Theorem 2.2, \(X_{n:n}\geq _{disp}Y_{n:n}\). This result has been acquired by [3] without the framework of Archimedean copulas.
It is also remarkable that in Theorem 2.2 the number \(m(x',\lambda )\) could be replaced by another finite upper bound of \(m(x,\lambda )\) such as \(m(\boldsymbol{\lambda} )\). That is, if one demonstrates that \(m(x,\boldsymbol{\lambda} ) \leq m(\boldsymbol{\lambda} )\), for all \(x\geq 0\), then the condition \(m(x',\boldsymbol{\lambda} ) \leq \lambda \) in Theorem 2.2 can be substituted by \(m(\boldsymbol{\lambda} ) \leq \lambda \). The following example illustrates another application of Theorem 2.2 in the dependent case.
Example 2.4
Consider the Clayton copula which has generator \(\phi (x)=(1+x)^{\frac{1}{\theta}}\), \(\theta >0\). We have \(\psi (x)=x^{\theta}1\). Let us take \(d(x)=\phi '(x)\) and recall that the conditions (i) and (ii) in Theorem 2.2 are satisfied as remarked before. Therefore,
Let us denote by \(\lambda _{(1)} \leq \cdots \leq \lambda _{(n)}\) the ordered values of \(\lambda _{i}\)’s, from the smallest to the biggest. Then, one obtains
Now, since for every \(j=1,2,\ldots , i\), \(\lambda _{(j)} \leq \lambda _{(i)}\), thus for all \(j=1,2,\ldots , i\) one has
and, consequently,
Hence, one has
Thus, as an application of Theorem 2.2,
The following example illustrates another application of Theorem 2.2.
Example 2.5
Let us consider Gumbel copula with generator \(\phi (x)=\exp (\sqrt{x})\). In the context of Theorem 2.2, we take \(d(x)=\phi (x)\). It can be observed after some routine calculation that \((1\phi (x))^{\frac{1}{d(x)}}\) is increasing in \(x>0\), and further that \(d(x)\frac{1\phi (x)}{\phi '(x)}\) is decreasing and convex in \(x>0\). Denote \(F_{\lambda _{i}}(x)=1e^{\lambda _{i} x}\). Now, we can get
where
By Lyapunov’s inequality, one obtains
For a fixed i, when \(i=1,2,\ldots ,n\), and by assuming without loss of generality that \(\lambda _{1}\leq \cdots \leq \lambda _{n}\), one thus gets
Since, for \(\lambda _{j} \geq \lambda _{i}\), \(\ln \left (\frac{F_{\lambda _{j}}(x)}{F_{\lambda _{i}}(x)}\right )\) is decreasing in \(x>0\), for every \(j>i\),
In parallel, \(\lambda _{j} \leq \lambda _{i}\), \(\ln \left (\frac{F_{\lambda _{j}}(x)}{F_{\lambda _{i}}(x)}\right )\) is increasing in \(x>0\), thus for every \(j\leq i\),
Therefore,
Hence, an application of Theorem 2.2 yields
The next example provides another application of Theorem 2.2.
Example 2.6
Let us take \(\psi (u)=(1u)^{\theta}\), \(\theta \geq 1\). In Theorem 2.2, let us take \(d(x)=1\). It can be verified that \((1\phi (x))^{\frac{1}{d(x)}}\) is increasing in \(x>0\), and also \(d(x)\frac{1\phi (x)}{\phi '(x)}\) is decreasing and convex in \(x>0\). Using Theorem 2.2, one gets
The following result also presents another set of conditions under which the dispersive order among the lifetimes of parallel systems with dependent exponential random variables is satisfied. The proof of it being similar to the proof of Theorem 2.2 has been omitted.
Theorem 2.7
Let \(\boldsymbol{X}=(X_{1},\ldots ,X_{n})\) be a dependent random vector with the Archimedean copula having generator ϕ where \(X_{i}\) has exponential distribution with hazard rate \(\lambda _{i}\) and, further, let \(\boldsymbol{Y}=(Y_{1},\ldots ,Y_{n})\) be another dependent random vector having the same Archimedean copula as X where \(Y_{i}\) has exponential distribution with hazard rate λ. Let there exist a nonnegative function d for which

(i)
\((1\phi (u))^{d(u)}\) is increasing (resp. decreasing) in \(u\geq 0\);

(ii)
\(d(u)\frac{1\phi (u)}{\phi '(u)}\) is increasing (resp. decreasing) and also concave in \(u\geq 0\).
If \(x''=\arg \min _{x\geq 0} m(x,\boldsymbol{\lambda} )\) and \(\lambda >0\), then
3 Usual stochastic order of parallel systems
Theorem 3.1
Let X be a dependent random vector with \(X_{i}\sim F(\cdot ;\lambda _{i})\) and with the Archimedean copula having generator ϕ and let Y be another dependent random vector with \(Y_{i}\sim F(\cdot ;\lambda _{i}^{*})\) and with the Archimedean copula having generator ϕ. Let

(i)
\(F(\cdot ;\lambda )\) be increasing in λ;

(ii)
\(F(\cdot ;\lambda )\) be logconcave in λ;

(iii)
\(u\psi '(u)\) be increasing in \(u\in (0,1)\).
Then, we have
Proof
The survival function of \(X_{n:n}\) is of the form
Thus for establishing the desired result, it is enough to show that the function \(\overline{F}_{X_{n:n}}(x;\boldsymbol{\lambda} )\) is Schurconvex. Taking the derivative of \(\overline{F}_{X_{n:n}}(x;\boldsymbol{\lambda} )\) with respect to \(\lambda _{i}\), we have
Therefore, for \(1\leq i< j\leq n\) and \(\boldsymbol{\lambda} \in \mathfrak{D}_{+}=\{\boldsymbol{\lambda}:\lambda _{1}\geq \lambda _{2}\geq \cdots \geq \lambda _{n}>0\}\), we show that \(\frac{\partial \overline{F}_{X_{n:n}}(x;\boldsymbol{\lambda} )}{\partial \lambda _{i}}\) is increasing in \(\lambda _{i}\). Thus, we can compute that
From the fact that \(\phi '\) is negative, it is enough to show that \(\eta _{1}\) is negative. We have
where \(\overset{sgn}{=}\) follows from the fact that \(\psi '\) is a negative function. The first inequality comes from the assumption that \(F(\cdot ;\lambda )\) is increasing in λ and \(u\psi '(u)\) is increasing in u, and the last inequality is based on the assumption \(F(\cdot ;\lambda )\) is logconcave. Then, the desired result follows by applying Lemma 1.3. For \(\boldsymbol{\lambda} \in \mathfrak{I}_{+}=\{\boldsymbol{\lambda}:0<\lambda _{1}\leq \lambda _{2}\leq \cdots \leq \lambda _{n}\}\), it can be proved in a similar manner that the desired result holds. □
Example 3.2
Consider the family of Lomax distributions with cdf \(F(x; \lambda )=\frac{\lambda x}{1+\lambda x}\), \(x\geq 0\), \(\lambda >0\), denoted by \(Lo(\lambda )\). Suppose that \(X_{i}\) follows \(Lo(\lambda _{i})\), with \(\lambda _{1}=2\), \(\lambda _{2}=3\), \(\lambda _{3}=4\) and, further, assume that \(Y_{i}\) follows \(Lo(\lambda ^{\star}_{i})\) with \(\lambda ^{\star}_{1}=1\), \(\lambda ^{\star}_{2}=3\), \(\lambda ^{\star}_{3}=5\). Contemplate the multivariate Gumbel copula having generator \(\psi _{\theta}(u)=(\ln (u))^{\theta}\), for \(\theta \geq 1\). It can be verified plainly that \(F(x; \lambda )\) is increasing and logconcave in λ, for all \(x \geq 0\). Moreover, \(u\psi '_{\theta}(u)\) is increasing in \(u\in (0,1)\). Since \(\boldsymbol{\lambda} \overset{m}{\preceq}\boldsymbol{\lambda} ^{*}\), using Theorem 3.1, we deduce that \(X_{3:3} \leq _{st} Y_{3:3}\). Note that
and
The graphs of \(\bar{F}_{X_{3:3}}(x;\boldsymbol{\lambda} )\) and \(\bar{F}_{Y_{3:3}}(x;\boldsymbol{\lambda} ^{\star})\) for \(\theta =2\) are plotted in Fig. 1 to acknowledge the achieved conclusion. The upper curve (blue) is the graph of \(\bar{F}_{Y_{3:3}}(x;\boldsymbol{\lambda} ^{\star})\) and the lower curve (red) is that of \(\bar{F}_{X_{3:3}}(x;\boldsymbol{\lambda} )\).
In the setting of Theorem 3.1, in order to see what happens if \(F(\cdot ;\lambda )\) is logconvex in \(\lambda >0\), a counterexample is presented below.
Counterexample 3.3
Suppose that \(F(x;\lambda )=\left (1e^{x^{3}}\right )^{\sqrt{\lambda }}\) where \(x>0\) and \(\lambda >0\). Denote by \(X_{1}\), \(X_{2}\), and \(X_{3}\) the component lifetimes of parallel system with dependent components, where \(X_{i} \sim F(\cdot ;\lambda _{i})\), \(i=1,2,3\) with Gumbel copula having parameter \(\theta >0\). Denote by \(Y_{1}\), \(Y_{2}\), and \(Y_{3}\) the component lifetimes of another parallel system with dependent components such that \(Y_{i} \sim F(\cdot ;\lambda ^{*}_{i})\) with the same Gumbel copula as that of \(X_{i}\)’s. It is easy to see that \(F(x;\lambda )\) is logconvex in λ, for all \(x>0\). Let us choose \(\boldsymbol{\lambda} =(4,5,6)\), \(\boldsymbol{\lambda} ^{*}=(3,5,7)\), and \(\theta =2\). Note that \(\boldsymbol{\lambda} \overset{m}{\preceq}\boldsymbol{\lambda} ^{*}\). It can be numerically verified that \(X_{3:3} \nleq _{st} Y_{3.3}\). Therefore, the result of Theorem 3.1 does not hold when \(F(\cdot ;\lambda )\) is logconvex in λ.
Proposition 3.4
Let X be a dependent random vector with \(X_{i}\sim F(\cdot ;\lambda _{i})\) and having the Archimedean copula with generator ϕ and let Y be another dependent random vector with \(Y_{i}\sim F(\cdot ;\lambda _{i}^{*})\) and having the Archimedean copula with generator ϕ. If

(i)
\(F(\cdot ;\lambda )\) is decreasing in λ;

(ii)
\(F(\cdot ;\lambda )\) is logconcave in λ;

(iii)
\(u\psi '(u)\) is decreasing in \(u\in (0,1)\).
Then, we have
Proof
In a manner similar to the proof of Theorem 3.1, we have that
where \(\eta _{1}=\frac{\partial{F}(x;\lambda _{j})}{\partial \lambda _{j}} \psi '(F(x,\lambda _{j}))  \frac{\partial{F}(x;\lambda _{i})}{\partial \lambda _{i}}\psi '(F(x, \lambda _{i})) \). It suffices to show that \(\eta _{1}\leq 0 \). Indeed, \(\eta _{1}\) can be rewritten as follows:
From the condition that \(F(\cdot ;\lambda )\) is logconcave in λ, it follows that
and from the assumption that \(u\psi '(u)\) is decreasing in \(u\in (0,1)\), one gets
By combining (5) and (6), it follows that \(\eta _{1}\) is nonpositive. Hence, the desired result follows by applying Lemma 1.3. □
It is notable that there are many Archimedean copulas satisfying the condition that \(u\psi '(u)\) increases or decreases in u of as required in Theorems 3.1 and 4.1. For example, if we consider the Ali–Mikhail–Haq (AMH) copula with the generator \(\phi (u)=\frac{1\theta}{e^{u}\theta}\) for \(\theta \in [1,1)\), it follows that \(\psi (u)=\ln \left (\frac{1\theta +\theta u}{u}\right )\). It can be seen that
which is increasing in u for \(\theta \in [0,1)\) and decreasing in u for \(\theta \in [1,0)\). For another example, consider Clayton copula with generator inverse \(\psi (u)=\frac{1}{\theta}( u^{\theta}1)\) for \(\theta \in [1,\infty )\setminus \{0\}\). It can be seen that
which is decreasing in u for \(\theta \in [1,0)\) and increasing in u for \(\theta \in [1,\infty )\).
Example 3.5
Consider the multivariate Clayton copula described by the generator \(\psi _{\theta}(x)=\frac{1}{\theta}\left (\frac{1}{x^{\theta}}1 \right )\), for \(\theta >0\) such that \(x\psi '_{\theta}(x)\) is increasing in \(x\in (0,1)\). Suppose that \(X_{i}\) and \(Y_{i}\), \(i=1,2,3\), are exponential random variables with density functions \(f_{X_{i}}(x)=\lambda _{i}e^{\lambda _{i}x}\), linked by an Archimedean copula with generator function \(\psi _{\theta}\) defined above and scale parameters \((\lambda _{1},\lambda _{2},\lambda _{3})=(2,4,5)\) and \((\lambda ^{*}_{1},\lambda ^{*}_{2},\lambda ^{*}_{3})=(1,3,7)\), respectively. It is easy to check that the conditions in Theorem 3.1 are satisfied.
In Proposition 3.4, to realize what happens if \(F(\cdot ;\lambda )\) is logconvex in \(\lambda >0\), the next counterexample is useful.
Counterexample 3.6
Let us take \(F(x;\lambda )=\left (\frac{x^{2}}{1+x^{2}}\right )^{\sqrt{\lambda }}\) where \(x,\lambda >0\). Let \(X_{1}\), \(X_{2}\), and \(X_{3}\) be three nonnegative dependent random variables, where \(X_{i} \sim F(\cdot ;\lambda _{i})\), \(i=1,2,3\) with AMH copula having parameter \(\theta \in [1,1)\). Further, let \(Y_{1}\), \(Y_{2}\), and \(Y_{3}\) be three nonnegative dependent random variables, where \(Y_{i} \sim F(\cdot ;\lambda ^{*}_{i})\) with the same AMH copula as that of \(X_{i}\)’s. It can be easily verified that \(F(x;\lambda )\) is logconvex in λ, for all \(x>0\). Suppose that \(\boldsymbol{\lambda} =(0.4,1.4,2.1)\), \(\boldsymbol{\lambda} ^{*}=(0.6,1.1,2.5)\), and let \(\theta =0.2\). We see that \(\boldsymbol{\lambda} \preceq _{w} \boldsymbol{\lambda} ^{*}\). Now, after some calculation, we observe that \(X_{3:3} \nleq _{st} Y_{3.3}\), and, consequently, Proposition 3.4 does not remain valid if \(F(\cdot ;\lambda )\) is logconvex in λ.
Theorem 3.7
Let X be a dependent random vector with \(X_{i}\sim F(\cdot ;\lambda _{i})\) and having the Archimedean copula with generator ϕ and let Y be another dependent random vector with \(Y_{i}\sim F(\cdot ;\lambda _{i}^{*})\) and having the Archimedean copula with generator ϕ. If

(i)
\(F(\cdot ;\lambda )\) is decreasing in λ;

(ii)
\(F(\cdot ;\lambda )\) is logconcave in λ;

(iii)
\(u\psi '(u)\) is decreasing in \(u\in (0,1)\).
Then, we have
Proof
By Theorem 3.1 and since \(F(\cdot ;\lambda )\) is decreasing in λ, we conclude that \(\overline{F}_{X_{n:n}}(x;\boldsymbol{\lambda} )\) is decreasing in \(\lambda _{i}\). For \(\boldsymbol{\lambda} ,\boldsymbol{\lambda} ^{*}\in \mathfrak{D}_{+}=\{\boldsymbol{\lambda}: \lambda _{1}\geq \lambda _{2}\geq \cdots \geq \lambda _{n}>0\}\), where \(\boldsymbol{\lambda} \preceq _{w}\boldsymbol{\lambda} ^{*}\), in accordance with Theorem 5.A.9 of [11], there exists some \(\boldsymbol{\beta} \in \mathfrak{D}_{+}\) such that \(\boldsymbol{\lambda} \preceq \boldsymbol{\beta} \) and \(\boldsymbol{\beta} \overset{m}{\preceq}\boldsymbol{\lambda} ^{*}\), where \(\boldsymbol{\lambda} \preceq \boldsymbol{\beta} \) means that \(\lambda _{i}\leq \beta _{i}\), \(i=1,\ldots ,n\). Under the assumptions \((i)\)–\((iii)\) of Proposition 3.4, it follows that \(\boldsymbol{\beta} \overset{m}{\preceq}\boldsymbol{\lambda} ^{*}\) implies \(T_{n:n}\leq _{st}Y^{*}_{n:n}\), where \(T_{n:n}\) is the lifetime of a parallel system formed from random lifetimes \(\boldsymbol{T}=(T_{1},\ldots ,T_{n})\) with \(T_{i}\sim F(\cdot ,\beta _{i})\) for \(i=1,\ldots ,n\). Also due to the assumption that \(\overline{F}_{X_{n:n}}(x;\boldsymbol{\lambda} )\) is increasing function in \(\lambda _{i} \), it follows that \(\boldsymbol{\lambda} \preceq \boldsymbol{\beta} \) implies \(\overline{F}_{X_{n:n}}(x;\boldsymbol{\lambda} )\leq \overline{F}_{T_{n:n}}(x; \boldsymbol{\beta} )\), which in turn implies that \(X_{n:n}\leq _{st}T_{n:n}\). By combining these observations, it holds that \(X_{n:n}\leq _{st}X^{*}_{n:n}\). For \(\boldsymbol{\lambda} \in \mathfrak{I}_{+}=\{\boldsymbol{\lambda}:0<\lambda _{1}\leq \lambda _{2}\leq \cdots \leq \lambda _{n}\}\), it can be proved in a similar manner that the desired result holds, hence the claim of the theorem. □
Example 3.8
Under the setup of Example 3.5, it is easy to check that \(x\psi '_{\theta}(x)\) is decreasing in x for all \(\theta \in [1,0)\). Suppose that \(X_{i}\) and \(Y_{i}\), \(i=1,2,3\), are exponential random variables with density functions \(f_{X_{i}}(x)=\frac{1}{\lambda _{i}}e^{\frac{1}{\lambda _{i}}x}\), linked by an Archimedean copula with generator function \(\psi _{\theta}\) and scale parameters \((\lambda _{1},\lambda _{2},\lambda _{3})=(1,2,4)\) and \((\lambda ^{*}_{1},\lambda ^{*}_{2},\lambda ^{*}_{3})=(2,3,5)\), respectively. It is easy to check that \((\lambda _{1},\lambda _{2},\lambda _{3})\preceq _{w}(\lambda ^{*}_{1}, \lambda ^{*}_{2},\lambda ^{*}_{3})\) and also the conditions in Theorem 3.7 are satisfied.
In the framework of Theorem 3.7, to see what happens if \(F(\cdot ;\lambda )\) is logconvex in \(\lambda >0\), we give the following counterexample.
Counterexample 3.9
Consider the cumulative distribution function \(F(x;\lambda )=e^{\frac{\sqrt{\lambda }}{x}}\) where \(x,\lambda >0\). We also consider \(X_{1}\), \(X_{2}\), and \(X_{3}\) as three nonnegative dependent random variables, where \(X_{i} \sim F(\cdot ;\lambda _{i})\), \(i=1,2,3\) with Clayton copula having parameter \(\theta >0\). Moreover, take \(Y_{1}\), \(Y_{2}\), and \(Y_{3}\) as three nonnegative dependent random variables, where \(Y_{i} \sim F(\cdot ;\lambda ^{*}_{i})\) with the same Archimedean copula as that of \(X_{i}\)’s. It can be seen that \(F(x;\lambda )\) is logconvex in λ, for all \(x>0\). Let \(\boldsymbol{\lambda} =(0.4,1.4,2.1)\), \(\boldsymbol{\lambda} ^{*} =(0.6,1.1,2.5)\), and \(\theta =4.25\). It is seen that \(\boldsymbol{\lambda} \preceq _{w} \boldsymbol{\lambda} ^{*}\). We observe that \(X_{3:3} \nleq _{st} Y_{3:3}\). Hence, the result of Theorem 3.7 is not satisfied in the case when \(F(\cdot ;\lambda )\) is logconvex in λ.
4 Usual stochastic order of series systems
This section develops the usual stochastic ordering of extreme order statistics in the dependent case provided that the original random variables follow a oneparameter lifetime distribution. We use the notation d.noni.d. in place of dependent and nonidentically distributed.
Theorem 4.1
Let \(\boldsymbol{X}=(X_{1},\dots ,X_{n})\) be a vector of nonnegative d.noni.d. random variables, where \(X_{i}\sim F(\cdot ;\lambda _{i})\) and the dependence structure follows the Archimedean copula with generator ϕ and let \(\boldsymbol{Y}=(Y_{1},\dots ,Y_{n})\) be another vector of d.noni.d. random variables following the Archimedean copula with generator ϕ, where \(Y_{i}\sim F(\cdot ;\lambda _{i}^{*})\). Let

(i)
\(\overline{F}(\cdot ;\lambda )\) be decreasing in λ;

(ii)
\(\overline{F}(\cdot ;\lambda )\) be logconvex in λ;

(iii)
\(u\psi '(u)\) be decreasing in \(u\in (0,1)\).
Then,
Proof
The distribution function of \(X_{1:n}\) is of the form
Thus for establishing the desired result, it is enough to show that the function \({F}_{X_{1:n}}(x;\boldsymbol{\lambda} )\) is Schurconcave. Set \(1\leq i< j\leq n\) and \(\boldsymbol{\lambda} \in \mathfrak{D}_{+}=\{\boldsymbol{\lambda}:\lambda _{1}\geq \lambda _{2}\geq \cdots \geq \lambda _{n}>0\}\). Taking the derivative of \({F}_{X_{1:n}}(x;\boldsymbol{\lambda} )\) with respect to \(\lambda _{i}\), we have
Therefore, for \(i\neq j\), it follows that
Due to the fact that \(\phi '\) is negative, it is enough to show that \(\eta _{2}\) is positive. We can compute
Since \(\overline{F}(\cdot ;\lambda )\) is logconvex with respect to λ, we have
and due to the condition that \(u\psi '(u)\) is decreasing in u and the fact that \(\overline{F}(x;\lambda _{i})\geq \overline{F}(x;\lambda _{j})\), we see that
By combining (7) and (9), it follows that \(\eta _{2}\) is nonnegative. From these observations, according to Lemma 1.3, it follows that \({F}_{X_{1:n}}\) is Schurconcave. This in turn guarantees that \(\overline{F}_{X_{1:n}}\) is Schurconvex. Hence, the desired result follows by applying Lemma 1.3. For \(\boldsymbol{\lambda} \in \mathfrak{I}_{+}=\{\boldsymbol{\lambda}:0<\lambda _{1}\leq \lambda _{2}\leq \cdots \leq \lambda _{n}\}\), it can be proved in a similar manner that the desired result holds. □
Example 4.2
Under the setup of Example 3.5, it is easy to check that \(x\psi '_{\theta}(x)\) is decreasing in x for all \(\theta \in [1,0)\) and that the conditions in Theorem 4.1 are satisfied.
In connection with Theorem 4.1, one can ask what happens if \(\bar{F}(\cdot ;\lambda )\) is logconcave in \(\lambda >0\). The following counterexample clarifies the issue.
Counterexample 4.3
Consider a parametric family of distributions with survival function \(\bar{F}(x;\lambda )=e^{\lambda x^{2}}\), \(x>0\), and \(\lambda >0\). Assume that \(X_{1}\), \(X_{2}\), and \(X_{3}\) are nonnegative dependent random variables, where \(X_{i} \sim \bar{F}(\cdot ;\lambda _{i})\), \(i=1,2,3\) with Gumbel copula having generator \(\phi (x)=e^{x^{\frac{1}{\theta }}}\), where \(\theta \geq 1\). Let \(Y_{1}\), \(Y_{2}\), and \(Y_{3}\) be nonnegative dependent random variables, where \(Y_{i} \sim \bar{F}(\cdot ;\lambda ^{*}_{i})\) with the same Archimedean copula as that of \(X_{i}\). It can be seen that \(\bar{F}(x;\lambda )\) is logconcave in λ, for all \(x>0\). Let us choose \(\boldsymbol{\lambda} =(3,4,5)\), \(\boldsymbol{\lambda} ^{*} =(2,4,6)\) which fulfill that \(\boldsymbol{\lambda} \overset{m}{\preceq}\boldsymbol{\lambda} ^{*}\). We consider Gumbel copula with \(\theta =3\). It can be seen that \(X_{1:3} \nleq _{st} Y_{1:3}\). As a result, Theorem 4.1 does not remain valid in situations where \(F(\cdot ;\lambda )\) is logconcave in λ.
Theorem 4.4
Let \(\boldsymbol{X}=(X_{1},\dots ,X_{n})\) be a vector of d.noni.d. random variables with \(X_{i}\sim F(\cdot ;\lambda _{i})\) and having the Archimedean copula with generator ϕ and let \(\boldsymbol{Y}=(Y_{1},\dots ,Y_{n})\) be another vector of d.noni.d. random variables with \(Y_{i}\sim F(\cdot ;\lambda _{i}^{*})\) and having the Archimedean copula with generator ϕ. If

(i)
\(\overline{F}(\cdot ;\lambda )\) is increasing in λ;

(ii)
\(\overline{F}(\cdot ;\lambda )\) is logconcave in λ;

(iii)
\(u\psi '(u)\) is increasing in \(u\in (0,1)\).
Then, we have
Proof
In a manner similar to the proof of Theorem 4.1 for establishing the desired result, it is enough to show that the function \({F}_{X_{1:n}}(x;\boldsymbol{\lambda} )\) is Schurconvex. Taking the derivative of \({F}_{X_{1:n}}(x;\boldsymbol{\lambda} )\) with respect to \(\lambda _{i}\), we have
Therefore, for \(i\neq j\), it follows that
Since \(\phi '\) is negative, it is enough to show that \(\eta _{2}\) is nonpositive. We can compute
where, using the same arguments as in proof of Theorem 4.1, the first inequality follows from the conditions that \(\overline{F}(x;\lambda _{i})\) is increasing in λ and \(u\psi '(u)\) is increasing in u. Since \(\psi '\) is nonpositive and \(\overline{F}(x;\lambda )\) is logconcave in λ, it holds that
from which the righthand side of (9) is nonpositive. Hence, the desired result follows by applying Lemma 1.3. □
In the next example, using Theorem 4.1, we present a situation where the reliability of a system with components of different ages used increases as the ages of the components become more widely separated. Recall that a nonnegative random variable X has a distribution with the \(IFR\) property when the hazard rate function of X, i.e., \(r_{X}(x)\), is increasing in \(x\geq 0\).
Example 4.5
Denote by X the lifetime of a fresh unit that has survival function F̄. Then the lifetime of the unit at age λ as the used unit, where \(\lambda > 0\) has survival function \(\bar{F}(x;\lambda )=\frac{\bar{F}(x+\lambda )}{\bar{F}(\lambda )}\). Assume that \(X_{i}\) is the lifetime of a used unit with age \(\lambda _{i}\), and further assume that \(Y_{i}\) is the lifetime of a used unit with age \(\lambda ^{\star}_{i}\), where \(\boldsymbol{\lambda} =(3,4,5)\) and \(\boldsymbol{\lambda} ^{*} =(2,4,6)\) are two quantities of different ages. If F possesses the \(IFR\) property with a concave hazard rate function, then it is easy to show that \(\bar{F}(x;\lambda )\) is decreasing in λ and is also logconvex in λ for all \(x \geq 0\). Consider the multivariate Gumbel copula with the generator \(\psi _{\theta}(u)=(\ln (u))^{\theta}\), for \(\theta \geq 1\). It can be proved that \(u\psi '_{\theta}(u)\) is increasing in \(u\in (0,1)\). Thus, since \(\boldsymbol{\lambda} \preceq _{m}\boldsymbol{\lambda} ^{*}\), using Theorem 4.1, one concludes that \(X_{1:3}\leq _{\mathrm{st}} Y_{1:3}\). Thus, considering two series systems with used components where the component lifetimes satisfy the \(IFR\) property and have a concave hazard rate function, the system comprising used components with close ages has a lower reliability compared to the system comprising used components with scattered ages in the sense of the majorization order.
In Theorem 4.4, we present a counterexample where \(\bar{F}(\cdot ;\lambda )\) is logconvex in \(\lambda >0\).
Counterexample 4.6
Let \(\bar{F}(x;\lambda )=e^{\sqrt{\lambda } x^{2}}\) be the underlying survival function where \(\lambda >0\). Suppose that \(X_{1}\), \(X_{2}\), and \(X_{3}\) are nonnegative dependent random variables, where \(X_{i} \sim \bar{F}(\cdot ;\lambda _{i})\), \(i=1,2,3\) with Gumbel copula having generator \(\phi (x)=e^{x^{\frac{1}{\theta }}}\), where \(\theta \geq 1\). In addition, we assume that \(Y_{1}\), \(Y_{2}\), and \(Y_{3}\) are nonnegative dependent random variables, where \(Y_{i} \sim \bar{F}(\cdot ;\lambda ^{*}_{i})\) with the same copula function as that of \(X_{i}\)’s. It can be seen that \(\bar{F}(x;\lambda )\) is logconvex in λ, for all \(x>0\). Let us choose \(\boldsymbol{\lambda} =(1,2,3)\), \(\boldsymbol{\lambda} ^{*} =(0.8,2,3.2)\), and take \(\theta =4.5\). It is observed that \(X_{1:3} \nleq _{st} Y_{1:3}\). It can be seen that \(\boldsymbol{\lambda} \overset{m}{\preceq}\boldsymbol{\lambda} ^{*}\). Therefore, the result of Theorem 4.4 does not hold in the case where \(\bar{F}(\cdot ;\lambda )\) is logconvex in λ.
5 Conclusion
In this study, the usual stochastic ordering between the lifetimes of two series/parallel systems with nonidentical lifetimes of the dependent components under an Archimedean copula dependence structure was performed under certain circumstances. It was assumed that the heterogeneous components follow a oneparameter lifetime distribution for which the cdf (e.g., \(F(x,\boldsymbol{\lambda} )\)) or the survival function (e.g., \(\bar{F}(x,\boldsymbol{\lambda} )\)) must satisfy the property of being a monotone function of λ which, moreover, is logconcave (logconvex) with respect to λ. The inverse generator of the Archimedean copula should satisfy some special properties. It was shown that the majorization order and the weak submajorization order are relevant to establish the usual stochastic order between the lifetimes of the systems. Several examples were used to investigate the sufficient conditions to obtain the results. In general, the problem of ordering the lifetimes of two parallel/series systems with respect to the usual stochastic ordering shows which system is more reliable, which can be a useful investigation in the context of decision making and optimization for the assembly of components in a system. In the study conducted in this paper, the selection of components with lifetimes that follow a general distribution with one parameter whose parameters are more scattered improves the reliability of the system compared to the case in which components are selected with lifetimes that follow the same family of distributions with less variable parameters.
In a future study, we could consider nonparametric distributions instead of the general family of distributions with one parameter associated with the component lifetimes in a parallel/series system when the component lifetimes are heterogeneous and, moreover, dependent with an Archimedean copula structure. In such a situation, it is an interesting development to establish the usual stochastic ordering between the lifetimes of two parallel/series systems. In addition, another study can be conducted to investigate the ordering properties of dependent random variables having a more general dependence structure than that of an Archimedean dependence structure.
Data Availability
No datasets were generated or analysed during the current study.
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The author is sincerely grateful to one anonymous reviewer for his/her constructive suggestions.
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This work was supported by Researchers Supporting Project number (RSP2024R464), King Saud University, Riyadh, Saudi Arabia.
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Shrahili, M. Stochastic ordering results on extreme order statistics from dependent samples with Archimedean copula. J Inequal Appl 2024, 120 (2024). https://doi.org/10.1186/s13660024032016
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DOI: https://doi.org/10.1186/s13660024032016