Some New Bivariate Properties and Characterizations Under Archimedean Copula

Abstract

This paper considers comparing properties and characterizations of the bivariate functions under Archimedean copula. It is shown that some results of the usual stochastic order for the bivariate functions in the independent case are generalized to the Archimedean copula-linked dependent case, and we also derive some characterizations of different bivariate functions composed by Archimedean copula-linked dependent random variables. These results generalize some existing results in the literature and bring conclusions closer to reality. Two applications in scheduling problems are also provided to illustrate the main results.

1. Introduction

Various concepts of stochastic ordering provide a variety of useful comparisons in insurance actuarial science, reliability, etc. Many researchers have studied the nature of the (reversed) hazard rate and likelihood ratio order. Lynch et al. [1] examined some closure properties of hazard rate order, while Oliveira and Torrado [2] showed the characteristics and closed properties of a decreasing proportional reversed hazard rate class. Boland et al. [3] presented the application of hazard rate order in reliability and [4] discussed the reliability application of the reversed hazard order. In the literature [5,6,7,8,9,10,11,12], the stochastic comparisons of series and parallel systems with independent components have been effectively investigated through the smallest and the largest order statistics in the sense of (reversed) hazard rate order. Esna-Ashari et al. [13] compared generalized order statistics in the sense of hazard rate order and reversed hazard order. Also, some scholars studied the likelihood ratio ordering properties of the smallest and the largest order statistics; see [14,15]. Furthermore, Barmalzan et al. [16] established the likelihood ratio order properties of the smallest claim amounts of two independent heterogeneous Weibull distributions.

On the other hand, the past several decades have witnessed many investigations on the bivariate characterization. Muraleedharan and Unnikrishnan [17] discussed the characterization of Gumbel’s bivariate exponential distribution based on the properties of the conditional moments. Shanthikumar and Yao [18] showed the bivariate characterization in the sense of the likelihood ratio order, the hazard rate order and the usual stochastic order. Meanwhile, based on collections of pairs of bivariate functions, Righter and Shanthikumar [19] extended the result in [18]. Thomas and Veena [20] derived some properties of concomitants of record values which characterize the generalized class of distributions. Noughabi and Kayid [21] proposed and studied bivariate $\alpha -$ quantile residual life measure.

Most of the existing research that compares random variables assumes that they are all independent. However, the operating environment of such technical systems is often affected by many factors, such as operating conditions, environmental conditions and the stress factors on the components. For this reason, it would be prudent to consider dependent random variables. Archimedean copulas are a type of multivariate probability distribution used to model the dependence relationship among random variables. Moreover, they also incorporate the independence copula as a special case. For a comprehensive review of copulas and their applications, see [22,23,24,25].

Under Archimedean copulas, Barmalzan et al. [26] proved the comparison properties of the hazard rate order and reversed hazard order, which is the extension of [8]. Li and Fang [27] discussed stochastic orderings of the extremal values and their adjacent order statistics in the sense of both likelihood ratio order and (reversed) hazard rate order, while Mesfioui et al. [28] extended the result of [27] to the heterogeneous random variables linked by an Archimedean copula. Ariyafar et al. [29] compared the aggregation and minimum of these portfolios with respect to the Laplace transform order according to Archimedean copulas. Fang et al. [30] presented the stochastic comparison results for the largest and smallest order statistics from dependent Gaussian variables with Archimedean copula under simple tree order restrictions. Lu et al. [31] used the asymmetric Archimedean copula to characterize the multiple uncertainties of a W-PV-H system. Amini et al. [32] considered two k-out-of-n systems comprising heterogeneous dependent components under random shocks with an Archimedean copula. Recently, Guan and Wang [33] proved stochastic comparison properties of convex and increasing convex order under Archimedean copulas. We find that for dependent random variables, fewer researchers have considered the usual stochastic order and the characterization of bivariate functions based on Archimedean copula. The objective of this study is to generalize the usual stochastic order and the characterization of bivariate functions composed by Archimedean copula-linked dependent random variables. The results on stochastic orders under Archimedean copulas mentioned above motivate us to consider the problems discussed in this work.

The rest of this paper is organized as follows. In Section 2, some basic definitions and notation are given which are most pertinent to the results established in the subsequent sections. In Section 3, some useful lemmas are provided which establish the relationship between independent and Archimedean copula-linked dependent random variables. In Section 4, some new results regarding the usual stochastic order and characterization among bivariate functions of dependent random variables are obtained, which extend some known results in the literature and bring conclusions closer to reality. In Section 5, two applications are given. Finally, we provide some conclusions in Section 6.

2. Preliminaries

Throughout this paper, we denote $\mathbb{R}=(-\infty ,\infty )$, ${\mathbb{R}}^{+}=[0,\infty )$, ${\mathbb{R}}^n=\{(z_1,\dots ,z_n):z_i\in (-\infty ,\infty )$, for all $i\}$. The terms “increasing” and “decreasing” are used to denote “monotone non-decreasing” and “monotone non-increasing”, respectively. We suppose that all random variables are defined on a common probability space $(\mathsf{\Omega },\mathcal{F},\mathit{P})$ and that all expectations exist wherever they are used.

2.1. Stochastic Orders

Definition 1.

Suppose that random variables X and Y have distribution functions $F_X$ and $F_Y,$ and survival functions ${\overline{F}}_X=1-F_X$ and ${\overline{F}}_Y=1-F_Y$, respectively.

 (i)

If ${\overline{F}}_X\left(x\right)\le {\overline{F}}_Y\left(x\right)$ for all $x\in \mathbb{R},$ then we say that X is smaller than Y in the usual stochastic order (denoted by $X{\le }_{st}Y$);

 (ii)

If ${\overline{F}}_Y\left(x\right)/{\overline{F}}_X\left(x\right)$ is increasing in $x\in {\mathbb{R}}^{+}$, then we say that X is smaller than Y in the hazard rate order (denoted by $X{\le }_{hr}Y$);

 (iii)

If $F_Y\left(x\right)/F_X\left(x\right)$ is increasing in $x\in {\mathbb{R}}^{+}$, then we say that X is smaller than Y in the reversed hazard order (denoted by $X{\le }_{rh}Y$);

 (iv)

If $f_Y\left(x\right)/f_X\left(x\right)$ is increasing in $x\in {\mathbb{R}}^{+}$, then we say that X is smaller than Y in the likelihood ratio order (denoted by $X{\le }_{lr}Y$).

It is well known that the likelihood ratio order implies the hazard rate order and the reversed hazard order, and the hazard rate order or the reversed hazard order implies the usual stochastic order. For more details on stochastic orderings and their applications, one may refer to [34,35].

2.2. Copulas

In this section, we present some basic notions of copulas.

Let F be an n-dimensional distribution function with marginal distributions $F_1,\dots ,F_n$. A copula associated with F is a distribution function $C:{[0,1]}^n\to [0,1]$ satisfying

$$F\left(x_1,\dots ,x_n\right)=C\left(F_1\left(x_1\right),\dots ,F_n\left(x_n\right)\right).$$

Similarly, a survival copula associated with $\overline{F}$ is a survival function $C:{[0,1]}^n\to [0,1]$ satisfying $\overline{F}\left(x_1,\dots ,x_n\right)=C\left({\overline{F}}_1\left(x_1\right),\dots ,{\overline{F}}_n\left(x_n\right)\right)$. For comprehensive discussions on copulas, one may refer to [36,37,38].

Definition 2

([36]). If for all $t\in (a,b)$, all derivatives of a function $h\left(t\right)$ exist and satisfy ${(-1)}^k{h}^{\left(k\right)}\left(t\right)\ge 0,k\in \{0,1,\dots \}$, where $h^{\left(k\right)}(·)$ denotes the kth derivative of $h(·)$; then, $h(·)$ is said to be completely monotone on an interval $(a,b),a,b\in \mathbb{R}$.

Thus, if ${\varphi }^{-1}$ is a completely monotone function, there exists a distribution function $L_{{\varphi }^{-1}}$ such that

$${\varphi }^{-1}\left(x\right)={\int }_0^{\infty }exp(-\alpha x)d{L}_{{\varphi }^{-1}}\left(\alpha \right).$$

Definition 3

([37]). For a completely monotone function ${\varphi }^{-1}:[0,+\infty )\to [0,1]$ with ${\varphi }^{-1}\left(0\right)=1$ and ${lim}_{x\to +\infty }{\varphi }^{-1}\left(x\right)=0$, then $C_{\varphi }\left(\mathit{u}\right)={\varphi }^{-1}\left(\varphi \left(u_1\right)+\cdots +\varphi \left(u_n\right)\right)$, for all $u_i\in (0,1),i\in I_n$, is called an Archimedean copula with strict generator ϕ, where ${\varphi }^{-1}\left(u\right)$ is the pseudo-inverse of ϕ.

For example, $C(u_1,\dots ,u_n)=min(u_1,\dots ,u_n)$ is an n-copula, and the Gumbel copula is a special case of strict Archimedean copulas with generator $\varphi \left(t\right)={(-lnt)}^{\theta },\theta \ge 1.$ For ease of notation, we denote the vector $\left(F_{X_1}\left(x_1\right),\dots ,F_{X_n}\left(x_n\right)\right)$ by ${\mathit{F}}_{{\mathit{X}}_n}\left({\mathit{x}}_n\right)$ and the vector $\left({\overline{F}}_{X_1}\left(x_1\right),\dots ,{\overline{F}}_{X_n}\left(x_n\right)\right)$ by ${\overline{\mathit{F}}}_{{\mathit{X}}_n}\left({\mathit{x}}_n\right)$.

3. Some Useful Lemmas

For establishing the main results in the following section, the following lemmas are required.

Lemma 1

([33]). Let $C\left({\overline{\mathit{F}}}_{{\mathit{U}}_n}\left({\mathit{u}}_n\right)\right)$ be the joint survival function of $(U_1,\dots ,U_n)$, where C is the Archimedean copula with generator ϕ. For $i=1,\cdots ,n$, let $U_i\left(\alpha \right)$’s be independent random variables with survival functions $exp\left\{-\alpha \varphi \left({\overline{F}}_{U_i}\right)\right\}$. Then, for all $\alpha >0$ and all continuous functions $g:{\mathbb{R}}^n\to \mathbb{R}$, we have

$$E\left[g\left(U_1,\dots ,U_n\right)\right]={\int }_0^{\infty }E\left[g\left(U_1\left(\alpha \right),\dots ,U_n\left(\alpha \right)\right)\right]d{L}_{{\varphi }^{-1}}\left(\alpha \right).$$

Remark 1.

In fact, the above conclusion only requires g to be a measurable function but not a continuous function. This is because the process of proving Lemma 1 mainly uses the Fubini theorem, which holds as long as g is a measurable function.

Lemma 2.

Let U and V be two random variables. Assume that $\varphi :(0,1]\to {\mathbb{R}}^{+}$ is a monotone decreasing function with $\varphi \left(1\right)=0$, $\varphi \left(0\right)=\infty$. For all $\alpha >0$, let the survival functions of $U\left(\alpha \right)$ and $V\left(\alpha \right)$ be given by $exp\left\{-\alpha \varphi \left({\overline{F}}_U\right)\right\}$ and $exp\left\{-\alpha \varphi \left({\overline{F}}_V\right)\right\}$, respectively. Then, $U\left(\alpha \right){\le }_{hr}V\left(\alpha \right)$ for all $\alpha >0$ if and only if $\varphi \left({\overline{F}}_V\left(t\right)\right)-\varphi \left({\overline{F}}_U\left(t\right)\right)$ is decreasing in $t\in {\mathbb{R}}^{+}$.

Proof. 

Note that for $\alpha >0$, we have

$$\begin{array}{ccc}\hfill U\left(\alpha \right){\le }_{hr}V\left(\alpha \right)& ⟺& {\displaystyle \frac{{\overline{F}}_{V\left(\alpha \right)}\left(t\right)}{{\overline{F}}_{U\left(\alpha \right)}\left(t\right)}}\mathrm{i}\mathrm{s}\mathrm{i}\mathrm{n}\mathrm{c}\mathrm{r}\mathrm{e}\mathrm{a}\mathrm{s}\mathrm{i}\mathrm{n}\mathrm{g}int\in {\mathbb{R}}^{+},\hfill \\ & ⟺& exp\{-\alpha [\varphi \left({\overline{F}}_V\left(t\right)\right)-\varphi \left({\overline{F}}_U\left(t\right)\right)]\}\mathrm{i}\mathrm{s}\mathrm{i}\mathrm{n}\mathrm{c}\mathrm{r}\mathrm{e}\mathrm{a}\mathrm{s}\mathrm{i}\mathrm{n}\mathrm{g}int\in {\mathbb{R}}^{+},\hfill \\ & ⟺& \varphi \left({\overline{F}}_V\left(t\right)\right)-\varphi \left({\overline{F}}_U\left(t\right)\right)\mathrm{i}\mathrm{s}\mathrm{d}\mathrm{e}\mathrm{c}\mathrm{r}\mathrm{e}\mathrm{a}\mathrm{s}\mathrm{i}\mathrm{n}\mathrm{g}int\in {\mathbb{R}}^{+},\hfill \end{array}$$

which is the desired result. □

Lemma 3.

Suppose U and V are two random variables. Assume that $\varphi :(0,1]\to {\mathbb{R}}^{+}$ is a monotone decreasing function with $\varphi \left(1\right)=0$, $\varphi \left(0\right)=\infty$. For all $\alpha >0$, let the survival functions of $U\left(\alpha \right)$ and $V\left(\alpha \right)$ be given by $exp\left\{-\alpha \varphi \left({\overline{F}}_U\right)\right\}$ and $exp\left\{-\alpha \varphi \left({\overline{F}}_V\right)\right\}$, respectively. Then, $U\left(\alpha \right){\le }_{rh}V\left(\alpha \right)$ for all $\alpha >0$ if and only if $\varphi \left(F_V\left(t\right)\right)-\varphi \left(F_U\left(t\right)\right)$ is decreasing in $t\in {\mathbb{R}}^{+}$.

Proof. 

For all $\alpha >0$,

$$\begin{array}{ccc}\hfill U\left(\alpha \right){\le }_{rh}V\left(\alpha \right)& ⟺& {\displaystyle \frac{F_{V\left(\alpha \right)}\left(t\right)}{F_{U\left(\alpha \right)}\left(t\right)}}\mathrm{i}\mathrm{s}\mathrm{i}\mathrm{n}\mathrm{c}\mathrm{r}\mathrm{e}\mathrm{a}\mathrm{s}\mathrm{i}\mathrm{n}\mathrm{g}\mathrm{i}\mathrm{n}t\in {\mathbb{R}}^{+}\hfill \\ & ⟺& exp\{-\alpha [\varphi \left(F_V\left(t\right)\right)-\varphi \left(F_U\left(t\right)\right)]\}\mathrm{i}\mathrm{s}\mathrm{i}\mathrm{n}\mathrm{c}\mathrm{r}\mathrm{e}\mathrm{a}\mathrm{s}\mathrm{i}\mathrm{n}\mathrm{g}\mathrm{i}\mathrm{n}t\in {\mathbb{R}}^{+}\hfill \\ & ⟺& \varphi \left(F_V\left(t\right)\right)-\varphi \left(F_U\left(t\right)\right)\mathrm{i}\mathrm{s}\mathrm{d}\mathrm{e}\mathrm{c}\mathrm{r}\mathrm{e}\mathrm{a}\mathrm{s}\mathrm{i}\mathrm{n}\mathrm{g}\mathrm{i}\mathrm{n}t\in {\mathbb{R}}^{+}.\hfill \end{array}$$

□

Lemma 4.

Suppose U and V are two random variables. Assume that $\varphi :(0,1]\to {\mathbb{R}}^{+}$ is a monotone decreasing function with $\varphi \left(1\right)=0$, $\varphi \left(0\right)=\infty$.

 (i)

For all $\alpha >0$, let the survival functions of $U\left(\alpha \right)$ and $V\left(\alpha \right)$ be given by $exp\left\{-\alpha \varphi \left({\overline{F}}_U\right)\right\}$ and $exp\left\{-\alpha \varphi \left({\overline{F}}_V\right)\right\}$, respectively. Then, $U\left(\alpha \right){\le }_{lr}V\left(\alpha \right)$ for all $\alpha >0$ if and only if

$${\displaystyle \frac{{\varphi }^{\prime }\left({\overline{F}}_V\right)}{{\varphi }^{\prime }\left({\overline{F}}_U\right)}}{\displaystyle \frac{f_V}{f_U}}exp\{\varphi \left({\overline{F}}_U\right)-\varphi \left({\overline{F}}_V\right)\}$$

is increasing in $t\in {\mathbb{R}}^{+}$.

 (ii)

For all $\alpha >0$, let the distribution functions of $U\left(\alpha \right)$ and $V\left(\alpha \right)$ be given by $exp\left\{-\alpha \varphi \left(F_U\right)\right\}$ and $exp\left\{-\alpha \varphi \left(F_V\right)\right\}$, respectively. Then, $U\left(\alpha \right){\le }_{lr}V\left(\alpha \right)$ for all $\alpha >0$ if and only if

$${\displaystyle \frac{{\varphi }^{\prime }\left(F_V\right)}{{\varphi }^{\prime }\left(F_U\right)}}{\displaystyle \frac{f_V}{f_U}}exp\{\varphi \left(F_U\right)-\varphi \left(F_V\right)\}$$

is increasing in $t\in {\mathbb{R}}^{+}$.

Proof. 

 (i)

For all $\alpha >0$,

$$\begin{array}{ccc}\hfill U\left(\alpha \right){\le }_{lr}V\left(\alpha \right)& ⟺& {\displaystyle \frac{f_{V\left(\alpha \right)}\left(t\right)}{f_{U\left(\alpha \right)}\left(t\right)}}\mathrm{i}\mathrm{s}\mathrm{i}\mathrm{n}\mathrm{c}\mathrm{r}\mathrm{e}\mathrm{a}\mathrm{s}\mathrm{i}\mathrm{n}\mathrm{g}\mathrm{i}\mathrm{n}t\in {\mathbb{R}}^{+}\hfill \\ & ⟺& {\displaystyle \frac{\alpha {\varphi }^{\prime }\left({\overline{F}}_V\right)f_{V}exp\{-\alpha \varphi \left({\overline{F}}_V\right)\}}{\alpha {\varphi }^{\prime }\left({\overline{F}}_U\right)f_{U}exp\{-\alpha \varphi \left({\overline{F}}_U\right)\}}}\mathrm{i}\mathrm{s}\mathrm{i}\mathrm{n}\mathrm{c}\mathrm{r}\mathrm{e}\mathrm{a}\mathrm{s}\mathrm{i}\mathrm{n}\mathrm{g}\mathrm{i}\mathrm{n}t\in {\mathbb{R}}^{+}\hfill \\ & ⟺& {\displaystyle \frac{{\varphi }^{\prime }\left({\overline{F}}_V\right)}{{\varphi }^{\prime }\left({\overline{F}}_U\right)}}{\displaystyle \frac{f_V}{f_U}}exp\{\varphi \left({\overline{F}}_U\right)-\varphi \left({\overline{F}}_V\right)\}\mathrm{i}\mathrm{s}\mathrm{i}\mathrm{n}\mathrm{c}\mathrm{r}\mathrm{e}\mathrm{a}\mathrm{s}\mathrm{i}\mathrm{n}\mathrm{g}\mathrm{i}\mathrm{n}t\in {\mathbb{R}}^{+}.\hfill \end{array}$$

 (ii)

The proof is similar to that of $\left(i\right)$.

□

4. Main Results

The following Theorem 1, which is a generalization of Theorem $1.B.9$ of [35], gives the usual stochastic ordering property of the bivariate functions under Archimedean copula.

Theorem 1.

Suppose $C\left({\overline{F}}_U\left(u\right),{\overline{F}}_V\left(v\right)\right)$ is the joint survival function of $(U,V)$, where C is the Archimedean copula with generator ϕ such that ${\varphi }^{-1}$ is a completely monotone function. Then, $\varphi \left({\overline{F}}_V\left(t\right)\right)-\varphi \left({\overline{F}}_U\left(t\right)\right)$ is decreasing in $t\in {\mathbb{R}}^{+}$ if and only if, for all $g\in {\mathcal{G}}_{hr}$,

$$g(U,V){\le }_{st}g(V,U).$$

where ${\mathcal{G}}_{hr}=\left\{g:{\mathbb{R}}^2\to \mathbb{R}:\right.$$g(u,v)$ is increasing in u, for each v, on $\{u\ge v\}$, and is decreasing in v, for each u, on $\{v\ge u\}$}.

Proof. 

For all $\alpha >0$, let the random variables $U\left(\alpha \right)$ and $V\left(\alpha \right)$ be independent, and let the survival function of $U\left(\alpha \right)$$\left(V\left(\alpha \right)\right)$ be $exp\{-\alpha \varphi \left({\overline{F}}_U\right)\}$$\left(exp\{-\alpha \varphi \left({\overline{F}}_V\right)\}\right)$, respectively. Note that the condition $\varphi \left({\overline{F}}_V\left(t\right)\right)-\varphi \left({\overline{F}}_U\left(t\right)\right)$ is decreasing in $t\in {\mathbb{R}}^{+}$; using Lemma 2, we have $U\left(\alpha \right){\le }_{hr}V\left(\alpha \right)$. Then, it can be obtained from Theorem $1.B.9$ of [35] that for all $g\in {\mathcal{G}}_{hr}$,

$$g(U\left(\alpha \right),V\left(\alpha \right)){\le }_{st}g(V\left(\alpha \right),U\left(\alpha \right)).$$

This means that for all increasing functions $\phi$, we have

$$E\left[\phi \left(g\left(U\left(\alpha \right),V\left(\alpha \right)\right)\right)\right]\le E\left[\phi \left(g\left(V\left(\alpha \right),U\left(\alpha \right)\right)\right)\right].$$

(1)

In addition, according to Lemma 1, we have

$$\begin{array}{c}\hfill E\left[\phi \left(g\left(U,V\right)\right)\right]={\int }_0^{\infty }E\left[\phi \left(g\left(U\left(\alpha \right),V\left(\alpha \right)\right)\right)\right]d{L}_{{\varphi }^{-1}}\left(\alpha \right),\end{array}$$

(2)

and

$$\begin{array}{c}\hfill E\left[\phi \left(g\left(V,U\right)\right)\right]={\int }_0^{\infty }E\left[\phi \left(g\left(V\left(\alpha \right),U\left(\alpha \right)\right)\right)\right]d{L}_{{\varphi }^{-1}}\left(\alpha \right).\end{array}$$

(3)

Based on (1)–(3), we can obtain

$$E\left[\phi \left(g\left(U,V\right)\right)\right]\le E\left[\phi \left(g\left(V,U\right)\right)\right].$$

Thus, we have $g(U,V){\le }_{st}g(V,U)$ for all $g\in {\mathcal{G}}_{hr}$.

Now, suppose for all $g\in {\mathcal{G}}_{hr}$,

$$g(U,V){\le }_{st}g(V,U).$$

Select a u and a v such that $u\ge v$. Let $g(x,y)=I_{\{x\ge u,y\ge v\}}$, where $I_A$ denotes the indicator function of the set A. It is easy to see that $g(x,y)$ is increasing in x. Additionally, for fixed x and y, such that $y\ge x$, we have that $g(x,y)=1$ if $x\ge u$ and $g(x,y)=0$ if $x<u$. Therefore, $g\in {\mathcal{G}}_{\mathrm{hr}}$. Hence, we have

$$Eg(V,U)\ge Eg(U,V),$$

(4)

whenever $u\ge v$. Note that

$$Eg(V,U)=E{I}_{\{V\ge u,U\ge v\}}={\varphi }^{-1}\left(\varphi \left({\overline{F}}_V\left(u\right)\right)+\varphi \left({\overline{F}}_U\left(v\right)\right)\right),$$

(5)

and

$$Eg(U,V)=E{I}_{\{U\ge u,V\ge v\}}={\varphi }^{-1}\left(\varphi \left({\overline{F}}_U\left(u\right)\right)+\varphi \left({\overline{F}}_V\left(v\right)\right)\right),$$

(6)

where ${\overline{F}}_U$ and ${\overline{F}}_V$ are the survival functions of U and V, respectively. Note that ${\varphi }^{-1}$ is a decreasing function; upon combining (4)–(6), we now have

$$\varphi \left({\overline{F}}_V\left(u\right)\right)-\varphi \left({\overline{F}}_U\left(u\right)\right)\le \varphi \left({\overline{F}}_V\left(v\right)\right)-\varphi \left({\overline{F}}_U\left(v\right)\right),$$

which means that $\varphi \left({\overline{F}}_V\left(t\right)\right)-\varphi \left({\overline{F}}_U\left(t\right)\right)$ is decreasing in $t\in {\mathbb{R}}^{+}$. Hence, the proof of the theorem is completed. □

Remark 2.

Theorem 1 shows a necessary and sufficient condition for bivariate functions consisting of two Archimedean copula-linked dependent random variables to maintain the usual stochastic order after exchanging the order of variables.

Theorem 2 below generalizes Theorem $1.B.47$ of [35] to an Archimedean copula-linked dependent case. The result shows another necessary and sufficient condition for bivariate functions to preserve a usual stochastic order under Archimedean copula.

Theorem 2.

Suppose $C\left({\overline{F}}_U\left(u\right),{\overline{F}}_V\left(v\right)\right)$ is the joint survival function of $(U,V)$, where C is the Archimedean copula with generator ϕ such that ${\varphi }^{-1}$ is a completely monotone function. For all $g\in {\mathcal{G}}_{rh}$, $\varphi \left(F_V\left(t\right)\right)-\varphi \left(F_U\left(t\right)\right)$ is decreasing in $t\in {\mathbb{R}}^{+}$ if and only if

$$g(U,V){\le }_{st}g(V,U),$$

where ${\mathcal{G}}_{rh}=\left\{g:{\mathbb{R}}^2\to \mathbb{R}:\right.$$g(u,v)$ is increasing in u, for each v, on $\{u\le v\}$, and is decreasing in v, for each u, on $\{v\le u\}$}.

Proof. 

For all $\alpha >0$, let $U\left(\alpha \right)$ and $V\left(\alpha \right)$ be independent random variables, and let the survival functions of $U\left(\alpha \right)$ and $V\left(\alpha \right)$ be given by $exp\left\{-\alpha \varphi \left({\overline{F}}_U\right)\right\}$ and $exp\left\{-\alpha \varphi \left({\overline{F}}_V\right)\right\}$, respectively. Note the condition that $\varphi \left({\overline{F}}_V\left(t\right)\right)-\varphi \left({\overline{F}}_U\left(t\right)\right)$ is decreasing in $t\in {\mathbb{R}}^{+}$; then, it can be obtained from Lemma 3 that $U\left(\alpha \right){\le }_{rh}V\left(\alpha \right)$. Then, according to Theorem $1.B.47$ of [35], for all $g\in {\mathcal{G}}_{rh}$, we have

$$g(U\left(\alpha \right),V\left(\alpha \right)){\le }_{st}g(V\left(\alpha \right),U\left(\alpha \right)),$$

which means that for any increasing function $\phi$,

$$E\left[\phi \left(g\left(U\left(\alpha \right),V\left(\alpha \right)\right)\right)\right]\le E\left[\phi \left(g\left(V\left(\alpha \right),U\left(\alpha \right)\right)\right)\right].$$

(7)

Moreover, by Lemma 1, we have

$$\begin{array}{c}\hfill E\left[\phi \left(g\left(U,V\right)\right)\right]={\int }_0^{\infty }E\left[\phi \left(g\left(U\left(\alpha \right),V\left(\alpha \right)\right)\right)\right]d{L}_{{\varphi }^{-1}}\left(\alpha \right),\end{array}$$

(8)

and

$$\begin{array}{c}\hfill E\left[\phi \left(g\left(V,U\right)\right)\right]={\int }_0^{\infty }E\left[\phi \left(g\left(V\left(\alpha \right),U\left(\alpha \right)\right)\right)\right]d{L}_{{\varphi }^{-1}}\left(\alpha \right).\end{array}$$

(9)

Using (7)–(9), we have

$$E\left[\phi \left(g\left(U,V\right)\right)\right]\le E\left[\phi \left(g\left(V,U\right)\right)\right],$$

which means

$$g(U,V){\le }_{st}g(V,U).$$

Now, suppose for all $g\in {\mathcal{G}}_{rh}$,

$$g(U,V){\le }_{st}g(V,U).$$

Select a u and a v such that $u\le v$. Let $g(x,y)=1-I_{\{x\le u,y\le v\}}$, where $I_A$ denotes the indicator function of the set A. It is easy to see that $g(x,y)$ is increasing in x. What is more, for fixed x and y, such that $y\le x$, we have that $g(x,y)=1$ if $x\ge u$ and $g(x,y)=0$ if $x<u$. Therefore, $g\in {\mathcal{G}}_{\mathrm{rh}}$. Hence, we have

$$Eg(V,U)\ge Eg(U,V),$$

(10)

whenever $u\le v$. Note that

$$Eg(V,U)=1-E{I}_{\{V\le u,U\le v\}}=1-{\varphi }^{-1}\left(\varphi \left(F_V\left(u\right)\right)+\varphi \left(F_U\left(v\right)\right)\right),$$

(11)

and

$$Eg(U,V)=1-E{I}_{\{U\le u,V\le v\}}=1-{\varphi }^{-1}\left(\varphi \left(F_U\left(u\right)\right)+\varphi \left(F_V\left(v\right)\right)\right),$$

(12)

where $F_U$ and $F_V$ are the distribution functions of U and V, respectively. Note that ${\varphi }^{-1}$ is a decreasing function; combining (10)–(12), we have

$$\varphi \left(F_V\left(u\right)\right)-\varphi \left(F_U\left(u\right)\right)\le \varphi \left(F_V\left(v\right)\right)-\varphi \left(F_U\left(v\right)\right),$$

which means that $\varphi \left(F_V\left(t\right)\right)-\varphi \left(F_U\left(t\right)\right)$ is decreasing in $t\in {\mathbb{R}}^{+}$. Thus, the required result follows. □

The following Theorem 3 generalizes Theorem $1.C.20$ of [35] to an Archimedean copula-linked dependent case. This result shows a sufficient condition for bivariate functions to preserve the usual stochastic order under Archimedean copula.

Theorem 3.

Let $C\left({\overline{F}}_U\left(u\right),{\overline{F}}_V\left(v\right)\right)$ be the joint survival function of $(U,V)$, where C is the Archimedean copula with generator ϕ such that ${\varphi }^{-1}$ is a completely monotone function. For all $g\in {\mathcal{G}}_{lr}=\left\{g:{\mathbb{R}}^2\to \mathbb{R}:g(u,v)\le g(v,u)\mathrm{whenever}u\le v\right\}$, if

$${\displaystyle \frac{{\varphi }^{\prime }\left({\overline{F}}_V\right)}{{\varphi }^{\prime }\left({\overline{F}}_U\right)}}{\displaystyle \frac{f_V}{f_U}}exp\{\varphi \left({\overline{F}}_U\right)-\varphi \left({\overline{F}}_V\right)\}$$

is increasing in $t\in {\mathbb{R}}^{+}$, then $g(U,V){\le }_{st}g(V,U)$.

Proof. 

For all $\alpha >0$, let the random variables $U\left(\alpha \right)$ and $V\left(\alpha \right)$ be independent, and let the survival functions of $U\left(\alpha \right)$ and $V\left(\alpha \right)$ be given by $exp\left\{-\alpha \varphi \left({\overline{F}}_U\right)\right\}$ and $exp\left\{-\alpha \varphi \left({\overline{F}}_V\right)\right\}$, respectively. Considering that ${\displaystyle \frac{{\varphi }^{\prime }\left({\overline{F}}_V\right)}{{\varphi }^{\prime }\left({\overline{F}}_U\right)}}{\displaystyle \frac{f_V}{f_U}}exp\{\varphi \left({\overline{F}}_U\right)-\varphi \left({\overline{F}}_V\right)\}$ is increasing in $t\in {\mathbb{R}}^{+}$, then, by Lemma 4, we have $U\left(\alpha \right){\le }_{lr}V\left(\alpha \right)$. Further, it can be acquired from Theorem $1.C.20$ of [35] that for all $g\in {\mathcal{G}}_{lr}$,

$$g(U\left(\alpha \right),V\left(\alpha \right)){\le }_{st}g(V\left(\alpha \right),U\left(\alpha \right)),$$

which means for all increasing functions $\phi$, we have

$$E\left[\phi \left(g\left(U\left(\alpha \right),V\left(\alpha \right)\right)\right)\right]\le E\left[\phi \left(g\left(V\left(\alpha \right),U\left(\alpha \right)\right)\right)\right].$$

(13)

In addition, using Lemma 1, we have

$$\begin{array}{c}\hfill E\left[\phi \left(g\left(U,V\right)\right)\right]={\int }_0^{\infty }E\left[\phi \left(g\left(U\left(\alpha \right),V\left(\alpha \right)\right)\right)\right]d{L}_{{\varphi }^{-1}}\left(\alpha \right),\end{array}$$

(14)

and

$$\begin{array}{c}\hfill E\left[\phi \left(g\left(V,U\right)\right)\right]={\int }_0^{\infty }E\left[\phi \left(g\left(V\left(\alpha \right),U\left(\alpha \right)\right)\right)\right]d{L}_{{\varphi }^{-1}}\left(\alpha \right).\end{array}$$

(15)

Combining (13)–(15), we obtain

$$E\left[\phi \left(g\left(U,V\right)\right)\right]\le E\left[\phi \left(g\left(V,U\right)\right)\right].$$

Therefore, for any $g\in {\mathcal{G}}_{lr}$, we have

$$g(U,V){\le }_{st}g(V,U).$$

This completes the proof. □

Similarly, we can prove the following result, which is the generalization of Theorem $1.C.23$ of [35] under Archimedean copula. The proof process is omitted for brevity.

Theorem 4.

Suppose $C\left({\overline{F}}_U\left(u\right),{\overline{F}}_V\left(v\right)\right)$ is the joint survival function of $(U,V)$, where C is the Archimedean copula with generator ϕ such that ${\varphi }^{-1}$ is a completely monotone function. If

$${\displaystyle \frac{{\varphi }^{\prime }\left({\overline{F}}_V\right)}{{\varphi }^{\prime }\left({\overline{F}}_U\right)}}{\displaystyle \frac{f_V}{f_U}}exp\{\varphi \left({\overline{F}}_U\right)-\varphi \left({\overline{F}}_V\right)\}$$

is increasing in $t\in {\mathbb{R}}^{+}$, then

$${\varphi }_1(U,V){\le }_{st}{\varphi }_2(U,V),$$

for all ${\varphi }_1$ and ${\varphi }_2$, that satisfy $\mathsf{\Delta }{\varphi }_{21}(u,v)\ge 0$ whenever $u\le v$, and ${\varphi }_1(u,v)\le$${\varphi }_2(v,u)$, for all u and v, where $\mathsf{\Delta }{\varphi }_{21}(u,v)={\varphi }_2(u,v)-{\varphi }_1(u,v).$

The following three theorems extend Theorem $1.B.10$, $1.B.48$ and $1.C.22$ of [35] to the Archimedean copula-linked dependent case. These results show three sufficient conditions for the characterization of different bivariate functions under Archimedean copula.

Theorem 5.

Let $C\left({\overline{F}}_U\left(u\right),{\overline{F}}_V\left(v\right)\right)$ be the joint survival function of $(U,V)$, where C is the Archimedean copula with generator ϕ such that ${\varphi }^{-1}$ is a completely monotone function. If $\varphi \left({\overline{F}}_V\left(t\right)\right)-\varphi \left({\overline{F}}_U\left(t\right)\right)$ is decreasing in $t\in {\mathbb{R}}^{+}$, then

$$E{\varphi }_1(U,V)\le E{\varphi }_2(U,V),$$

for all ${\varphi }_1$ and ${\varphi }_2$ such that, $\mathsf{\Delta }{\varphi }_{21}(u,v)$ increases in v on $\{v\ge u\}$, and such that $\mathsf{\Delta }{\varphi }_{21}(u,v)\ge -\mathsf{\Delta }{\varphi }_{21}(v,u)$ whenever $u\le v$, where $\mathsf{\Delta }{\varphi }_{21}(u,v)={\varphi }_2(u,v)-{\varphi }_1(u,v)$.

Proof. 

For all $\alpha >0$, let $U\left(\alpha \right)$ and $V\left(\alpha \right)$ be independent random variables, and let the survival function of $U\left(\alpha \right)$$\left[V\right(\alpha \left)\right]$ be $exp\{-\alpha \varphi \left({\overline{F}}_U\right)\}$$[exp\{-\alpha \varphi \left({\overline{F}}_V\right)\}]$, respectively. Since $\varphi \left({\overline{F}}_V\left(t\right)\right)-\varphi \left({\overline{F}}_U\left(t\right)\right)$ is decreasing in $t\in {\mathbb{R}}^{+}$, by Lemma 2, we have $U\left(\alpha \right){\le }_{hr}V\left(\alpha \right)$. Further, it then can be obtained from Theorem $1.B.10$ of [35] that

$$E{\varphi }_1(U\left(\alpha \right),V\left(\alpha \right))\le E{\varphi }_2(U\left(\alpha \right),V\left(\alpha \right)).$$

(16)

Furthermore, by Lemma 1, we obtain

$$\begin{array}{c}\hfill E{\varphi }_1(U,V)={\int }_0^{\infty }E{\varphi }_1(U\left(\alpha \right),V\left(\alpha \right))d{L}_{{\varphi }^{-1}}\left(\alpha \right),\end{array}$$

(17)

and

$$\begin{array}{c}\hfill E{\varphi }_2(U,V)={\int }_0^{\infty }E{\varphi }_2(U\left(\alpha \right),V\left(\alpha \right))d{L}_{{\varphi }^{-1}}\left(\alpha \right).\end{array}$$

(18)

Using (16)–(18), we obtain

$$E{\varphi }_1(U,V)\le E{\varphi }_2(U,V),$$

and completing the proof. □

Remark 3.

Theorem 5 indicates a sufficient condition for the expectation of different bivariate functions composed of two Archimedean copula-linked dependent random variables to maintain the size relationship.

Theorem 6.

Suppose $C\left({\overline{F}}_U\left(u\right),{\overline{F}}_V\left(v\right)\right)$ is the joint survival function of $(U,V)$, where C is the Archimedean copula with generator ϕ such that ${\varphi }^{-1}$ is a completely monotone function. If $\varphi \left(F_V\left(t\right)\right)-\varphi \left(F_U\left(t\right)\right)$ is decreasing in $t\in {\mathbb{R}}^{+}$, then

$$E{\varphi }_1(U,V)\le E{\varphi }_2(U,V),$$

for all ${\varphi }_1$ and ${\varphi }_2$ such that for each v, $\mathsf{\Delta }{\varphi }_{21}(u,v)$ decreases in u on $\{u\le v\}$ and such that $\mathsf{\Delta }{\varphi }_{21}(u,v)\ge -\mathsf{\Delta }{\varphi }_{21}(v,u)$ whenever $u\le v$, where $\mathsf{\Delta }{\varphi }_{21}(u,v)={\varphi }_2(u,v)-{\varphi }_1(u,v)$.

Theorem 7.

Let $C\left({\overline{F}}_U\left(u\right),{\overline{F}}_V\left(v\right)\right)$ be the joint survival function of $(U,V)$, where C is the Archimedean copula with generator ϕ such that ${\varphi }^{-1}$ is a completely monotone function. For all functions ${\varphi }_1$ and ${\varphi }_2$, $\mathsf{\Delta }{\varphi }_{21}(u,v)\ge 0$ whenever $u\le v$, and $\mathsf{\Delta }{\varphi }_{21}(u,v)\ge -\mathsf{\Delta }{\varphi }_{21}(v,u)$ whenever $u\le v$, $\mathsf{\Delta }{\varphi }_{21}(u,v)={\varphi }_2(u,v)-{\varphi }_1(u,v)$, if

$${\displaystyle \frac{{\varphi }^{\prime }\left({\overline{F}}_V\right)}{{\varphi }^{\prime }\left({\overline{F}}_U\right)}}{\displaystyle \frac{f_V}{f_U}}exp\{\varphi \left({\overline{F}}_U\right)-\varphi \left({\overline{F}}_V\right)\}$$

is increasing in $t\in {\mathbb{R}}^{+}$, then $E{\varphi }_1(U,V)\le E{\varphi }_2(U,V).$

Remark 4.

The proofs of both Theorems 6 and 7 are similar to the proof of Theorem 5; for the sake of brevity, they are omitted here.

Remark 5.

It should be emphasized that the results in Theorems 3, 4 and 7 still hold when $C\left({\overline{F}}_U\left(u\right),{\overline{F}}_V\left(v\right)\right)$ is replaced by $C\left(F_U\left(u\right),F_V\left(v\right)\right)$ and the condition that the increasing function ${\displaystyle \frac{{\varphi }^{\prime }\left({\overline{F}}_V\right)}{{\varphi }^{\prime }\left({\overline{F}}_U\right)}}{\displaystyle \frac{f_V}{f_U}}exp\{\varphi \left({\overline{F}}_U\right)-\varphi \left({\overline{F}}_V\right)\}$ is replaced by ${\displaystyle \frac{{\varphi }^{\prime }\left(F_V\right)}{{\varphi }^{\prime }\left(F_U\right)}}{\displaystyle \frac{f_V}{f_U}}exp\{\varphi \left(F_U\right)-\varphi \left(F_V\right)\}$.

5. Scheduling Application

In this section, two potential applications of our results are presented. Consider the following scheduling problem. In order to minimize the total cost, n jobs are scheduled on one machine. The processing time $X_i$ of the ith job and the processing time $X_j$ of the jth job are linked by an Archimedean copula with generator $\varphi$. $D_i$ denotes the completion time of job i, $g_i$ denotes its cost function; then, $TC={\sum }_{i=1}^n{g}_i\left(D_i\right)$ denotes the total cost. If $g_i-g_j$ is increasing, $g_i$ is said to be steeper than $g_j$, which is written $g_i{\ge }_s{g}_j$. Policy $\pi$ schedules job j immediately following job i, while policy ${\pi }^{*}$ is the same as $\pi$ except for interchanging jobs i and j. Under these settings, the following two theorems, which are obtained by using Theorems 4 and 5, can be used to compare these two totals.

Theorem 8.

Suppose $C\left({\overline{F}}_{X_i}\left(x_i\right),{\overline{F}}_{X_j}\left(x_j\right)\right)$ is the joint survival function of $(X_i,X_j)$, where C is the Archimedean copula with generator ϕ such that ${\varphi }^{-1}$ is a completely monotone function. If

$${\displaystyle \frac{{\varphi }^{\prime }\left({\overline{F}}_{X_j}\right)}{{\varphi }^{\prime }\left({\overline{F}}_{X_i}\right)}}{\displaystyle \frac{f_{X_j}}{f_{X_i}}}exp\{\varphi \left({\overline{F}}_{X_i}\right)-\varphi \left({\overline{F}}_{X_j}\right)\}$$

is increasing in $t\in {\mathbb{R}}^{+}$, $g_i$ is increasing for all i, and $g_i{\ge }_s{g}_j$ for all i and j, then

$$T{C}_{\pi }{\le }_{st}T{C}_{{\pi }^{*}}.$$

Theorem 9.

Suppose $C\left({\overline{F}}_{X_i}\left(x_i\right),{\overline{F}}_{X_j}\left(x_j\right)\right)$ is the joint survival function of $(X_i,X_j)$, where C is an Archimedean copula with generator ϕ such that ${\varphi }^{-1}$ is a completely monotone function. If $\varphi \left({\overline{F}}_{X_j}\left(t\right)\right)-\varphi \left({\overline{F}}_{X_i}\left(t\right)\right)$ is decreasing in $t\in {\mathbb{R}}^{+}$, $g_i$ is increasing for all i, and $g_i{\ge }_s{g}_j$ for all i and j, then

$$E\left(T{C}_{\pi }\right)\le E\left(T{C}_{{\pi }^{*}}\right).$$

6. Concluding Remarks

Suppose that U and V are two random variables. Furthermore, for all $\alpha >0$, let $U\left(\alpha \right)$ and $V\left(\alpha \right)$ be two independent random variables such that the survival (distribution) functions of $U\left(\alpha \right)$ and $V\left(\alpha \right)$ are $exp\left\{-\alpha \varphi \left({\overline{F}}_U\right)\right\}$ ($exp\left\{-\alpha \varphi \left(F_U\right)\right\}$) and $exp\left\{-\alpha \varphi \left({\overline{F}}_V\right)\right\}$$(exp\left\{-\alpha \varphi \left(F_V\right)\right\})$, respectively. Then, for all $\alpha >0$, we have obtained the equivalence relationship between the (reversed) hazard rate order (likelihood ratio order) of $U\left(\alpha \right)$ and $V\left(\alpha \right)$ and the functions of U and V. These equivalence relationships play a key role in deriving the results of this paper.

Using these equivalence relationships, two sufficient and necessary conditions for functions of two dependent random variables to preserve usual stochastic order were provided. We have proved two sufficient conditions for the functions of two dependent random variables based on the usual stochastic order. And we have also shown three sufficient conditions for the characterization of different bivariate functions composed by Archimedean copula-linked dependent random variables. These results generalize some existing results in the literature and bring conclusions closer to reality. Two applications in scheduling problems are also provided to illustrate the main results.

Due to the importance of stochastic order comparisons for dependent random variables, it is worth considering that the copulas in each set of the random variables are different. We shall study this problem and plan to report these results in future work.