Ordering and ageing properties of developed sequential order statistics governed by the Archimedean copula

2. Preliminaries and useful lemmas

Unless otherwise stated, we use the following notation throughout the paper. For an absolutely continuous random variable X, we denote the probability density function, the cumulative distribution function, the quantile function, the survival function, the hazard function, the reversed hazard function, the mean residual function, the cumulative hazard rate function, and the cumulative reversed hazard rate function by $f_X$ , $F_X$ , $F^{-1}_X$ , $\bar F_X$ , $r_X$ , $\tilde r_X$ , $m_X$ , ${\Delta}_X$ , and $\tilde \Delta_X$ , respectively; here $r_X\equiv f_X/\bar F_X$ , $\tilde r_X\equiv f_X/ F_X$ , $\Delta_X\equiv -\!\ln \bar F_X$ , $\tilde \Delta_X\equiv -\!\ln F_X$ , and $m_X(t)={\int\limits_t^\infty \bar F_X(w)dw}/{\bar F_X(t)}, \text{ for } t\geq 0.$

We use the following acronyms throughout the paper. We write ‘DID’, ‘OS’, ‘SOS’, and ‘DSOS’ for the phrases ‘dependent and identically distributed’, ‘ordinary order statistic(s)’, ‘sequential order statistic(s)’, and ‘developed sequential order statistic(s)’, respectively. By ‘ $\stackrel{d}=$ ’ we mean equality in distribution. All random variables considered in this paper are assumed to be absolutely continuous with strictly increasing cumulative distribution functions.

Copulas are a very effective tool for describing the dependency structure between random variables. In the literature, a large variety of copulas have been introduced to describe different dependency structures. Some of the best-known copulas are the Farlie–Gumbel–Morgenstern copula, the extreme-value copula, the family of Archimedean copulas, and the Clayton–Oakes copula. Among all of these, the family of Archimedean copulas has received the most attention from the researchers because of its mathematical tractability and its ability to describe a wide range of dependency structures. For an encyclopedic treatment of this topic, one may refer to Nelsen [Reference Nelsen33]. Below we give the definition of an Archimedean copula (see [Reference McNeil and Néslehová30]).

Definition 2.1. Let $\phi\;:\; [0, +\infty]\longrightarrow [0,1]$ be a decreasing continuous function such that $\phi(0)=1$ and $\phi(+\infty)=0$ , and let $\psi\equiv\phi^{-1}$ be the pseudo-inverse of $\phi$ . Then

(1) \begin{eqnarray} C(u_1,u_2,\dots,u_n)=\phi\!\left(\psi(u_1)+\psi(u_2)+\dots +\psi(u_n)\right),\quad \text{for }(u_1,u_2,\dots,u_n)\in [0,1]^n, \;\;\;\;\; \end{eqnarray}

is called the Archimedean copula with generator $\phi$ if $(\!-\!1)^k\phi^{(k)}(x)\geq 0$ , for $k=0,1,\dots,n-2$ , and $(\!-\!1)^{n-2}\phi^{(n-2)}(x)$ is decreasing and convex in $x\geq 0$ . $\Box$

As an extension of SOS, Baratnia and Doostparast [Reference Baratnia and Doostparast3] introduced the notion of DSOS, which is useful for describing the lifetimes of systems with dependent components. Below we describe the notion of DSOS governed by the Archimedean copula (see [Reference Baratnia and Doostparast3, Reference Navarro and Burkschat31]).

Let $F_1,F_2, \dots ,F_n$ be n absolutely continuous cumulative distribution functions with $F_1^{-1} (1)\leq \dots \leq F_n^{-1}(1)$ . Consider a system of n components installed at time $t = 0$ . Assume that all components of the system are functioning at the time of inception. Let $X^{(1)}_1, X^{(1)}_2, \dots , X^{(1)}_n$ be n DID random variables, with the distribution function $F _1$ , representing the lifetimes of the n components. Assume that the dependency structure between these random variables is described by the Archimedean copula with generator $\phi$ . Then the first component failure time is given by

\begin{align*} X^{\star}_{1:n} = \min\!\Big\{X_1^{(1)},X_2^{(1)}, \dots, X_n^{(1)}\Big\}.\end{align*}

Given $X^{\star}_{1:n} = t_1$ , the residual lifetimes of the $(n - 1)$ remaining components are equal in distribution to the residual lifetimes of $(n-1)$ DID components with age $t_1$ and with the distribution function $F_2 $ (instead of $F_1$ ) with the same dependency structure; here $F_2$ is assumed in place of $F_1$ because the failure of the first component has an impact on the performance of the other components. Let the lifetimes of these DID components be represented by $X^{(2)}_1, X^{(2)}_2, \dots , X^{(2)}_{n-1}$ . Then $X^{(2)}_j \sim F_2(\cdot | t_1)$ , where $\bar{F}_2(x|t_1) = {\bar{F}_2(x)}/{\bar{F}_2(t_1)}$ for $x \geq t_1$ . Moreover, $X^{(2)}_j \geq t_1$ , for $j = 1,2, \dots ,n - 1$ . Furthermore, the second component failure time is given by

\begin{align*} X^{\star}_{2:n} = \min\!\Big\{X_1^{(2)},X_2^{(2)}, \dots, X_{n-1}^{(2)}\Big\}.\end{align*}

Proceeding in this manner, we assume that the ith failure occurs at time $t_i \;(>t_{i-1})$ , i.e., $X^{\star}_{i:n} = t_i$ . Then the residual lifetimes of the $(n - i)$ remaining components are equal in distribution to the residual lifetimes of $(n-i)$ DID components with age $t_i$ and the distribution function $F_{i+1}$ with the same dependency structure. Let the lifetimes of these DID components be represented by $X^{(i+1)}_1, X^{(i+1)}_2, \dots , X^{(i+1)}_{n-i}$ . Then $X^{(i+1)}_j \sim F_{i+1}(\cdot | t_i)$ , where $\bar{F}_{i+1}(x|t_i) = {\bar{F}_{i+1}(x)}/{\bar{F}_{i+1}(t_i)}$ for $x \geq t_i$ . Moreover, note that $X^{(i+1)}_j \geq t_i$ , for $j = 1,2, \dots ,n - i$ . The $(i+1)$ th component failure time is then given by

\begin{align*} X^{\star}_{i+1:n} = \min\!\Big\{X_1^{(i+1)},X_2^{(i+1)}, \dots, X_{n-i}^{(i+1)}\Big\}.\end{align*}

Finally, if the $(n - 1)$ th component failure occurs at time $t_{n-1} = X^{\star}_{n-1:n}$ , then the last component failure time is given by $X^{\star}_{n:n}$ with the reliability function $\bar{F}_{n}(x|t_{n-1}) = {\bar{F}_{n}(x)}/{\bar{F}_{n}(t_{n-1})}$ for $x \geq t_{n-1}$ . The random variables $X^{\star}_{1:n} \leq X^{\star}_{2:n} \leq \dots \leq X^{\star}_{n:n} $ are called the developed sequential order statistics (DSOS) based on $F_1, F_2, \dots, F_n$ , where the dependency structure is described by the Archimedean copula with generator $\phi$ . For brevity, we denote them by $\big(X^{\star}_{1:n}, X^{\star}_{2:n}, \dots,X^{\star}_{n:n} \big)\sim$ DSOS( $ F_1 , F_2 , \dots , F_n;\;\phi $ ).

Remark 2.2. One may note that if $(X^{\star}_{1:n}, X^{\star}_{2:n}, \dots,X^{\star}_{n:n} )\sim$ DSOS( $ F_1 , F_2 , \dots , F_n;\;\phi $ ), then $\{X^{\star}_{1:n}, X^{\star}_{2:n},\dots ,X^{\star}_{n:n}\}$ forms a Markov chain with transition probabilities given by

(2) \begin{eqnarray}P\big(X^{\star}_{r:n} > t | X^{\star}_{r-1:n} =x\big) = \phi\bigg(\!(n-r+1) \psi \bigg( \frac{\bar{F}_r(t)}{\bar{F}_r(x)}\bigg)\!\bigg), \quad t\geq x>0,\end{eqnarray}

where $\bar{F} (x) >0$ and $\psi\equiv\phi^{-1}$ . $\Box$

Below we give an alternative definition of DSOS (see [Reference Cramer and Kamps19]).

Definition 2.2. Let $F_1, \dots, F_n$ be cumulative distribution functions with $F_1^{-1} (1)\leq \dots \leq F_n^{-1}(1)$ , and let

\begin{align*} \Big(Y_j^{(r)}\Big)_{1 \leq r \leq n, 1 \leq j \leq n-r+1}\end{align*}

be dependent random variables with $Y_j^{(r)} \sim F_r$ , $r= 1, 2, \dots, n$ , $j=1, 2, \dots, n-r+1$ , where the dependency structures are described by the same Archimedean copula with generator $\phi$ . Let $X_j^{(1)} = Y_j^{(1)}$ , $j=1,2, \dots, n$ , and $X^{\star}_{1:n} =\min\!\Big\{X_1^{(1)},X_2^{(1)}, \dots, X_n^{(1)}\Big\}.$ For $r=2, 3, \dots, n$ , let

\begin{align*} X_j^{(r)} = F_r^{-1} \Big\{F_r\Big(Y_j^{(r)}\Big)\big[1- F_r\big(X^{\star}_{r-1: n}\big)\big]+ F_r\big(X^{\star}_{r-1: n}\big)\Big\}, \quad j= 1, 2, \dots, n-r+1,\end{align*}

and $X^{\star}_{r:n} = \min\!\Big\{X_1^{(r)},X_2^{(r)}, \dots, X_{n-r+1}^{(r)}\Big\}$ . Then

\begin{align*} \big(X^{\star}_{1:n}, X^{\star}_{2:n}, \dots,X^{\star}_{n:n} \big)\sim \text{DSOS}( F_1 , F_2 , \dots , F_n;\;\phi ).\end{align*}

Two equivalent representations of DSOS are given in the following two lemmas. The proofs of these lemmas can be carried out along the same lines as in [Reference Burkschat and Navarro13, Reference Cramer and Kamps19] and are therefore omitted.

Lemma 2.1. Let $\big(X^{\star}_{1:n}, X^{\star}_{2:n}, \dots,X^{\star}_{n:n} \big)\sim$ DSOS( $ F_1 , F_2 , \dots , F_n;\;\phi $ ). Then

\begin{eqnarray*} &&X_{1:n}^{\star} = \bar{F}_1^{-1} \Big(V^{(1)}\Big), \\[5pt] && X_{i:n}^{\star} = \bar{F}_i^{-1}\Big(V^{(i)} \bar{F}_i\big(X_{i-1:n}^{\star}\big)\Big), \quad\text{ for } i= 2, 3, \dots, n, \end{eqnarray*}

where

\begin{align*} V^{(i)}= \max \Big\{\Big(1-U^{(i)}_1\Big), \dots , \Big(1-U^{(i)}_{n-i+1}\Big)\Big\}\end{align*}

and $U_j^{(i)} \sim Unif(0,1)$ , for $i=1,2 ,\dots, n$ and $j=1, 2, \dots, n-i+1$ , and, for each $i\in\{1,2,\dots, n\}$ , the $U_j^{(i)}$ are dependent random variables governed by the Archimedean copula with generator $\phi$ .

Lemma 2.2. Let $\big(X^{\star}_{1:n}, X^{\star}_{2:n}, \dots,X^{\star}_{n:n} \big)\sim$ DSOS( $ F_1 , F_2 , \dots , F_n;\;\phi $ ). Furthermore, let $D_i\equiv -\!\ln\bar{F}_i$ be the cumulative hazard rate function of $F_i$ , for $i=1,2,\dots,n$ . Then

(3) \begin{eqnarray} X_{1:n}^{\star} = D_1^{-1}\Big(W^{(1)}\Big), \end{eqnarray}

(4) \begin{eqnarray} X_{i:n}^{\star} = D_i^{-1}\Big(W^{(i)} + D_i\big(X_{i-1:n}^{\star}\big)\Big), \quad\text{ for } i=2,3, \dots, n, \end{eqnarray}

where

\begin{align*} W^{(i)} = -\!\ln\!\Big(V^{(i)}\Big)= \min\!\Big\{-\!\ln\Big(1-U^{(i)}_1\Big), \dots , -\!\ln \!\Big(1-U^{(i)}_{n-i+1}\Big)\Big\}, \quad i=1,2,\dots, n,\end{align*}

and the $U_j^{(i)}$ are the same as in Lemma 2.1. Moreover, $\{W^{(j)},\; j=1,2, \dots , n\}$ are independent random variables with

(5) \begin{eqnarray} \bar{F}_{W^{(j)}} (t) = \phi\big( (n-j+1) \psi \big( e^{-t}\big) \big), \quad t>0,\;j=1,2, \dots , n, \end{eqnarray}

where $\psi\equiv\phi^{-1}$ . $\Box$ .

Stochastic orders are widely used to compare two or more random variables/vectors. In the literature, numerous types of stochastic orders have been introduced, e.g., the usual stochastic order, the hazard rate order, etc. (see [Reference Shaked and Shanthikumar36, Reference Sahoo and Hazra37]). Below we give the definitions of several stochastic orders that are used in subsequent sections.

Like stochastic orders, stochastic ageings are also very useful tools for describing how a system behaves over time. In the literature, numerous ageing classes (e.g., IFR, IFRA, DFR, DLR, etc.) have been introduced to characterize different ageing properties of a system (see [Reference Barlow and Proschan4, Reference Lai and Xie27, Reference Sengupta and Deshpande38] and the references therein). Below we give the definitions of some ageing classes that are used in this paper.

Below we give a list of lemmas that are used in the next section. The proofs of Lemmas 2.4, 2.5, and 2.6 are omitted for the sake of brevity.

The proportional hazard rate model (PHR) model is one of the commonly used semi-parametric models. This model has many applications in survival analysis, reliability theory, and many other fields (see [Reference Marshall and Olkin29]). A set of random variables $\{Z_1,Z_2,\dots,Z_n\}$ is said to follow the PHR model if, for $i=1,2,\dots,n$ ,

\begin{align*} \bar F_{Z_i}(t)=(\bar F(t))^{\alpha_i}, \text{ for some }\alpha_i>0\;\;\text{and for all }t>0,\end{align*}

where $\bar F$ is the baseline survival function. We denote this PHR model by $F_{Z_i}\sim$ PHR( $ F;\; \alpha_i$ ), for $i=1,2\dots,n$ .

3. Main results

In this section we discuss the main results of this paper. First we give some stochastic comparison results for DSOS. We consider both one-sample and two-sample scenarios. Furthermore, we study some ageing properties of DSOS. In what follows, we introduce some notation.

Let $\big(X^{\star}_{1:n}, X^{\star}_{2:n}, \dots,X^{\star}_{n:n} \big)\sim$ DSOS( $ F_1 , F_2 , \dots , F_n;\;\phi $ ). For $i = 1, 2, \dots , n - 1$ , let

\begin{eqnarray*}Y^{(i+1)}_{1:n-i}= \min\!\Big\{Y^{(i+1)}_1, Y^{(i+1)}_2, \dots , Y^{(i+1)}_{n-i}\Big\},\end{eqnarray*}

where $Y^{(i+1)}_1, Y^{(i+1)}_2, \dots , Y^{(i+1)}_{n-i}$ are DID random variables with the distribution function $ F_{i+1}$ and the dependency structure described by the Archimedean copula with generator $\phi$ . Here, $Y^{(i+1)}_{k}$ is the random variable corresponding to the parent distribution of the kth remaining component at the ith step (i.e., after the ith failure), for $k=1,2,\dots,n-i$ . For the sake of convenience, we call this the kth parent random variable at the ith step. Consequently, $Y^{(i+1)}_{1:n-i}$ represents the minimum order statistic of all parent random variables at the ith step. Intuitively, what this means is as follows. Suppose that all surviving components at the ith step are replaced by a set of new components (i.e., with age zero) with lifetimes having the same distributions as the remaining surviving components have, i.e., $F_{i+1}$ . Then $Y^{(i+1)}_{1:n-i}$ represents the first failure time for this set of new components.

Furthermore, for an Archimedean copula with the generator $\phi$ , we use the following notation:

\begin{eqnarray*}&& H(u)=\frac{u\phi'(u)}{1-\phi(u)},\;\;R(u)=\frac{u\phi'(u)}{\phi (u)}\text{ and }G(u)=\frac{u\phi''(u)}{\phi'(u)},\quad u>0.\end{eqnarray*}

Note that $H(\!\cdot\!)$ , $R(\!\cdot\!)$ and $G(\!\cdot\!)$ are all negative-valued functions, because $\phi(\!\cdot\!)$ is a decreasing and convex function.

4. Examples

In this section, we discuss some examples to demonstrate the sufficient conditions given in the theorems of the previous section. Note that these sufficient conditions are satisfied by many popular Archimedean copulas (with specific choices of parameters), including the Clayton copula

\begin{align*} C(u_1,u_2,\dots,u_n) =\Bigg(\prod\limits_{i=1}^{n} u_i^{-\theta} - n + 1\Bigg)^{-1/\theta}\end{align*}

with the generator $\phi(t) = (\theta t + 1)^{-1/\theta}$ , for $\theta \geq 0$ , the Ali–Mikhail–Haq (AMH) copula

\begin{align*} C(u_1,u_2,\dots,u_n) =\Bigg((1 - \theta)\prod\limits_{i=1}^{n} u_i\Bigg)/\Bigg(\prod\limits_{i=1}^{n}(1- \theta +\theta u_i) -\theta \prod\limits_{i=1}^{n} u_i\Bigg)\end{align*}

with the generator $\phi(t) = (1 - \theta)/(e^t-\theta)$ , for $\theta \in [0, 1)$ , and the Gumbel–Hougaard copula

\begin{align*} C(u_1,u_2,\dots,u_n)=\exp\Bigg(-\Bigg[\sum\limits_{i=1}^n(-\!\ln u_i)^\theta\Bigg]^{1/\theta}\Bigg)\end{align*}

with the generator $\phi(t)=\exp\!\left(-t^{1/\theta}\right)$ , for $\theta \in[1,\infty)$ , among others. For the sake of completeness, below we give six examples. More examples can be found in [Reference Sahoo and Hazra37].

The first four examples illustrate the condition given in Parts (a) and (d) of Theorem 3.1, Theorem 3.2(a), Theorem 3.3(a), Theorem 3.5(a), Parts (a), (c), and (d) of Theorem 3.6, Theorem 3.7, and Theorem 3.8. The first two examples are borrowed from [Reference Sahoo and Hazra37].

Example 4.1. Consider the Archimedean copula with generator

\begin{eqnarray*} \phi(u) = e^{-u^{\frac{1}{\delta_1}}},\quad\delta_1 \in\!\left[1,\infty\right), \; u>0,\end{eqnarray*}

which gives

\begin{eqnarray*} \frac{uR'(u)}{R(u)} = \frac{1}{\delta_1}, \quad u>0.\end{eqnarray*}

It is trivially true that ${uR'(u)}/{R(u)}$ is positive and increasing in $u>0$ . Thus, the required condition is satisfied.

Example 4.2. Consider the Archimedean copula with generator

\begin{eqnarray*} \phi(u) = e^{1-(1+u)^{\frac{1}{\delta_2}}},\quad \delta_{2} \in\!\left(0,\infty\right), \; u>0.\end{eqnarray*}

From this, we have

\begin{eqnarray*} R(u) = -\frac{1}{\delta_2}u(1+u)^{\frac{1}{\delta_2}-1},\quad u>0,\end{eqnarray*}

and

\begin{eqnarray*} \frac{uR'(u)}{R(u)} = 1+\bigg(\frac{1}{\delta_2}-1\bigg)\frac{u}{u+1}, \quad u>0.\end{eqnarray*}

Let us fix $0 < \delta_{2} \leq 1$ . It can easily be shown that ${uR'(u)}/{R(u)}$ is positive and increasing in $u>0$ . Thus, the required condition is satisfied.

Example 4.3. Consider the Archimedean copula with generator

\begin{eqnarray*} \phi(u) = 1-(1-e^{-u})^{\frac{1}{\delta_{3}}},\quad \delta_{3} \in\!\left[1,\infty\right), \; u>0,\end{eqnarray*}

which gives

\begin{eqnarray*} \frac{uR'(u)}{R(u)} = 1-u+\frac{(1-\delta_{3})ue^{-u}}{\delta_{3}(1-e^{-u})}+\frac{ue^{-u}(1-e^{-u})^{\frac{1}{\delta_{3}}-1}}{\delta_{3}\Big(1-(1-e^{-u})^{\frac{1}{\delta_{3}}}\Big)}, \; u>0.\end{eqnarray*}

Writing $n_{1}(u,\delta_3)=uR'(u)/R(u)$ and $n_{2}(u,\delta_3)={\partial }/{\partial u}(uR'(u)/R(u))$ , $u>0$ , $\delta_3 \in [1, 10]$ , we plot $n_{1}(\!-\!\ln\!(v), \delta_3)$ and $n_{2}(\!-\!\ln\!(v), \delta_3)$ against $ (v, \delta_3) \in(0,1] \times [1, 10]$ . From Figures 1a and 1b, we see that $n_{1}(\!-\!\ln\!(v), \delta_3)$ and $n_{2}(\!-\!\ln\!(v), \delta_3)$ are positive in $ (v, \delta_3) \in(0,1] \times [1, 10]$ , and hence $uR'(u)/R(u)$ is positive and increasing in $u>0$ for $1 \leq \delta_3 \leq 10 $ . Thus, the required condition is satisfied.

Figure 1. Plots of $n_1$ , $n_2$ , $n_3$ , $n_4$ , $n_5$ , $n_6$ , $n_7$ , and $n_8$ .

Example 4.4. Consider the Archimedean copula with generator

\begin{eqnarray*} \phi(u) = -\frac{1}{\delta_4}\ln\!\big(1+ e^{-u}\big(e^{-\delta_4}-1\big) \big),\quad \delta_{4} \in\big(\!-\!\infty,\infty\big)\setminus \{0\}, \; u>0,\end{eqnarray*}

which gives

\begin{eqnarray*} \frac{uR'(u)}{R(u)} = \frac{ u - e^{\delta_4} u + (1 - e^{\delta_4} - e^{u + \delta_4} (\!-\!1 + u)) \ln\!\left(1 + e^{-u} ( e^{-\delta_4} -\!1)\right)}{\!\left(1 - e^{\delta_4} + e^{u + \delta_4}\right) \ln\!\left(1 + e^{-u} ( e^{-\delta_4} -\!1 )\right) }, \; u>0.\end{eqnarray*}

Writing $n_{3}(u,\delta_4)=uR'(u)/R(u)$ and $n_{4}(u,\delta_4)={\partial }/{\partial u}(uR'(u)/R(u))$ , $u>0$ , $\delta_4 \in [\!-\!40, -30]$ , we plot $n_{3}(\!-\!\ln\!(v), \delta_4)$ and $n_{4}(\!-\!\ln\!(v), \delta_4)$ against $ (v, \delta_4) \in(0,1] \times [-40, -30]$ . From Figures 1c and 1d, we see that $n_{3}(\!-\!\ln\!(v), \delta_4)$ and $n_{4}(\!-\!\ln\!(v), \delta_4)$ are positive in $ (v, \delta_3) \in(0,1] \times [\!-\!40, -30]$ , and hence $uR'(u)/R(u)$ is positive and increasing in $u>0$ for $-40 \leq \delta_4 \leq -30 $ . Thus, the required condition is satisfied.

The next two examples demonstrate the condition given in Theorem 3.1(b), Theorem 3.2(b), Theorem 3.3(b), Theorem 3.5(b), and Theorem 3.6(b).

Example 4.5. Consider the Archimedean copula with generator

\begin{eqnarray*} \phi(u) = e^{-u^{\frac{1}{\delta_5}}},\quad\delta_5 \in\!\left[1,\infty\right), \; u>0,\end{eqnarray*}

which gives

\begin{eqnarray*} \frac{uH'(u)}{H(u)} = \frac{1}{\delta_5}\Big(1-u^{\frac{1}{\delta_5}}\Big)-\frac{u^{\frac{1}{\delta_5}}e^{-u^{\frac{1}{\delta_5}}}}{\delta_5\Big(1-e^{-u^{\frac{1}{\delta_5}}}\Big)}, \quad u>0. \end{eqnarray*}

Writing $n_{5}(u,\delta_5)=uH'(u)/H(u)$ and $n_{6}(u,\delta_5)={\partial }/{\partial u}(uH'(u)/H(u))$ , $u>0$ , $\delta_5 \in [13, 19]$ , we plot $n_{5}(\!-\!\ln\!(v), \delta_5)$ and $n_{6}(\!-\!\ln\!(v), \delta_5)$ against $ (v, \delta_5) \in(0,1] \times [13, 19]$ . From Figures 1e and 1f, we see that $n_{5}(\!-\!\ln\!(v), \delta_5)$ and $n_{6}(\!-\!\ln\!(v), \delta_5)$ are negative in $ (v, \delta_5) \in(0,1] \times [13, 19]$ , and hence $uH'(u)/H(u)$ is negative and decreasing in $u>0$ for $13 \leq \delta_5 \leq 19 $ . Thus, the required condition is satisfied.

Example 4.6. Consider the Archimedean copula with generator

\begin{eqnarray*} \phi(u) = \!\left(\delta_6 u +1\right)^{-\frac{1}{\delta_6}}, \quad\delta_6 \in\!\left[-1,\infty\right)\setminus \{0\}, \; u>0,\end{eqnarray*}

which gives

\begin{eqnarray*} \frac{uH'(u)}{H(u)} = \frac{-1 + \!\left(\delta_6 u +1\right)^{\frac{1}{\delta_6}}- u\!\left(\delta_6 u +1\right)^{\frac{1}{\delta_6}} }{\big(\delta_6 u +1\big) \bigg( \big(\delta_6 u +1\big)^{\frac{1}{\delta_6}} -1\bigg)}, \quad u>0. \end{eqnarray*}

Writing $n_{7}(u,\delta_6)=uH'(u)/H(u)$ and $n_{8}(u,\delta_6)={\partial }/{\partial u}(uH'(u)/H(u))$ , $u>0$ , $\delta_6 \in [0.2, 0.9]$ , we plot $n_{7}(\!-\!\ln\!(v), \delta_6)$ and $n_{8}(\!-\!\ln\!(v), \delta_6)$ against $ (v, \delta_6) \in(0,1] \times [0.2, 0.9]$ . From Figures 1g and 1h, we see that $n_{7}(\!-\!\ln\!(v), \delta_5)$ and $n_{8}(\!-\!\ln\!(v), \delta_6)$ are negative in $ (v, \delta_6) \in(0,1] \times [0.2, 0.9]$ , and hence $uH'(u)/H(u)$ is negative and decreasing in $u>0$ for $0.2 \leq \delta_6 \leq 0.9 $ . Thus, the required condition is satisfied.

Below we cite two examples that illustrate the condition given in Theorem 3.1(c), Theorem 3.2(c), Theorem 3.3(c), and Theorem 3.5(c).

Example 4.7. Consider the Archimedean copula with generator

\begin{eqnarray*} \phi(u) = e^{\frac{1}{\delta_7}\!\left(1-e^u\right)}, \quad \delta_7 \in\!\left(0,1\right],\;u>0. \end{eqnarray*}

Then

\begin{eqnarray*} \frac{uG'(u)}{G(u)} = \frac{\delta_7-e^u-ue^u}{\delta_7-e^u},\quad \frac{G(u)}{R(u)}=1-\frac{\delta_7}{e^u}, \quad u>0, \end{eqnarray*}

and

\begin{eqnarray*} n_9(u)\stackrel{\text{def.}}=\frac{\partial }{\partial u}\bigg(\frac{uG'(u)}{G(u)}\bigg) = \frac{e^u\left(e^u-\delta_7(1+u)\right)}{\!\left(\delta_7-e^u\right)^2}, \quad u>0. \end{eqnarray*}

It can easily be shown that ${uG'(u)}/{G(u)}$ is positive, $n_9(u)$ is positive, and ${G(u)}/{R(u)}$ is increasing in $u>0$ . Thus, ${uG'(u)}/{G(u)}$ is positive and increasing and ${G(u)}/{R(u)}$ is increasing in $u>0$ . Consequently, the required condition holds from Remark 3.1.

Example 4.8. Consider the Archimedean copula with generator

\begin{eqnarray*} \phi(u) = e^{1-\!\left(1+u\right)^{\frac{1}{\delta_{8}}}},\quad \delta_{8} \in\!\left(0,\infty\right) \; u>0. \end{eqnarray*}

From this, we have

\begin{eqnarray*} &&G(u)=-\frac{1}{\delta_{8}}u\!\left(1+u\right)^{\frac{1}{\delta_{8}}-1}+u\!\left(1+u\right)^{-1}\bigg(\frac{1}{\delta_{5}}-1\bigg),\quad u>0, \\[5pt] &&R(u)=-\frac{1}{\delta_{8}}u\!\left(1+u\right)^{\frac{1}{\delta_{8}}-1},\quad u>0, \\[5pt] && \frac{G(u)}{R(u)}=1-\frac{1-\delta_{8}}{\!\left(1+u\right)^{\frac{1}{\delta_{8}}}}, \quad u>0. \end{eqnarray*}

Writing $n_{10}(u,\delta_8)=uG'(u)/G(u)$ and $n_{11}(u,\delta_8)={\partial }/{\partial u}(uG'(u)/G(u))$ , $u>0$ , $\delta_8 \in [0.5, 0.9]$ , we plot $n_{10}(\!-\!\ln\!(v), \delta_8)$ and $n_{11}(\!-\!\ln\!(v), \delta_8)$ against $ (v, \delta_8) \in(0,1] \times [0.5, 0.9]$ . From Figures 2a and 2b, we see that $n_{10}(\!-\!\ln\!(v), \delta_8)$ and $n_{11}(\!-\!\ln\!(v), \delta_8)$ are positive in $ (v, \delta_8) \in(0,1] \times [0.5, 0.9]$ , and hence $uG'(u)/G(u)$ is positive and increasing in $u>0$ for $0.5 \leq \delta_8 \leq 0.9 $ . Furthermore, it can easily be shown that $G(u)/R(u)$ is increasing in $u>0$ for $0.5 \leq \delta_8 \leq 0.9 $ . Thus, the required condition is satisfied. Consequently, the required condition holds from Remark 3.1.

Figure 2. Plots of $n_{10}$ and $n_{11}$ .

5. Concluding remarks

Most systems used in real life are very complex in nature; consequently, the components of these systems are interdependent, and the failure of one component affects the performance of the remaining working components. One effective way to model these systems is by using developed sequential order statistics (DSOS). In this paper, we study ordering and ageing properties of DSOS, and discuss some numerical examples. Our study generalizes many well-known results that are available for generalized order statistics and ordinary order statistics.

The main idea of this paper is to study systems with dependent components where dependency structures are modeled by Archimedean copulas. Among all existing copulas, the family of Archimedean copulas is the most popular one because of its mathematical tractability and ability to capture a wide spectrum of dependency structures. Thus, the proposed ordering results for DSOS, governed by the Archimedean copula, may be useful for comparing failure times of different components of a given system. Moreover, we can use these results to compare the lifetimes of two or more systems with dependent components in a given scenario. The proposed results on stochastic ageings may be helpful for understanding how a system ages as time progresses.

In this paper, some of the results are developed for a specific model. For example, all of the results given in Theorem 3.5 are derived for the proportional hazard rate model. The study of the same problem in a general setup (i.e., with arbitrary cumulative distribution functions) would be an interesting topic for future work.