Further results on geometric and harmonic mean failure rates, the associated aging intensity orders and classes of lifetime distribution

Time-to-failure of a device is in fact a non-negative random variable which follows a lifetime distribution. Analysis of lifetime distributions have been useful and constructive in the context of reliability and survival analysis (see, for instances, Lai and Xie [12] and Lawlesss [13]). There are several quantities of interest in reliability which measure the aging behaviour of lifetime distributions (cf. Finkelstein [6]). For example, the hazard rate function measures the instantaneous risk of failure of an alive organism or system and in contrast to it, the reversed hazard rate function quantifies the risk of failure of an inactive system right after its inactivity observation. Let X be a non-negative random variable (possibly denotes the time-to-failure of a device) with an absolutely continuous distribution function \(F_{X}\) and probability density function \(f_{X}\). The survival function of X is defined by \(\bar{F}_{X}\equiv 1-F_{X}\). The hazard rate function of X at time point t is a reliability quantity that measures the chance of instantaneous risk of failure of a used device of age t exactly at the time t. The second-hand devices which are still working at time t can be considered. We may consider the hazard rate function as the conditional probability of the failure of the device at age t, given that it did not fail before the age t. Specifically, for all \(t\geq 0\) for which \(\bar{F}_{X}(t)>0\), the failure (or hazard) rate function of X is defined as The most common use of this function is to model an entity’s chance of death as a function of their age. Roy and Mukherjee [18] considered different means of failure rate on the interval \((0,t)\) including arithmetic mean, geometric mean and harmonic mean. They used these means as new reliability measures to characterize some specific probability distributions and defined some new classes of lifetime distributions. As is known, the failure rate function measure the instantaneous risk of failure at a present time and relates only to the risk of immediate failure. However, the functions constructed using means of failure rate may prove to be more relevant than the failure rate function as they summarize the associated lifetime distribution at the entire time. The arithmetic mean failure rate of X (denoted by \(A_{X}\)), the geometric mean failure rate of X (denoted by \(G_{X}\)) and the harmonic mean failure rate of X (denoted by \(H_{X}\)) are defined as follows:

Recently, Bhattacharjee et al. [5] defined weighted failure rate and their means from the stand point of an application. The aging intensity function is also a novel reliability measure for assessing the aging phenomenon. Jiang et al. [7] have pointed out that a unimodal failure rate can be effectively viewed as either approximately decreasing or approximately increasing or approximately constant. It is clear that such representation is qualitative. In their paper, a quantitative measure, called aging intensity (AI) function, defined as the ratio of the failure rate to a baseline failure rate, has been studied. One choice of the baseline failure rate could be the average failure rate. The aging intensity function of X is defined as \(L_{X}(t):=\frac{h_{X}(t)}{\frac{1}{t}\int _{0}^{t} h_{X}(x) dx}\) with \(t>0\) (see, e.g., Nanda et al. [15]). In the AI function, the expression in the denominator is the arithmetic mean of hazard rate function. Bhattacharjee et al. [3] analyzed the aging classes based on monotonicity of new functions viz., arithmetic mean, geometric mean, and harmonic mean of the AI function. Szymkowiak et al. [21] then introduced generalized version of aging functions based on arithmetic, geometric and harmonic means of generalized aging intensity function. Recently, Bhattacharjee et al. [4] introduced some new reliability measures based on novel versions of aging intensity function applying alternative means of the failure rate function in the denominator. They, defined the harmonic aging intensity (HAI) function and the geometric aging intensity (GAI) function of X as and respectively. Bhattacharjee et al. [4] obtained some results involving the measures proposed in (3) and (4). They obtained these measures for some well-known parametric distributions and as a result they examined some classes of lifetime distribution constructed using these measures.

Three kinds of aging intensity function provide an overall measure of aging across a population or system. They average the aging characteristics over time, which can smooth out irregularities and provide a clearer picture of long-term trends. The harmonic, geometric, and the usual aging intensity functions serve as vital tools in reliability theory and survival analysis. They enhance our understanding of how systems and populations age, allowing for better decision-making in maintenance, healthcare, finance, and engineering. By quantifying aging processes, these functions aid in optimizing performance, improving safety, and enhancing overall efficiency across various fields.

In this section we give some necessary preliminaries as we need in the sequel. Definitions of several stochastic orders as well as some related classes of lifetime distributions are given. The following definition is due to Bhattacharjee et al. [4] which introduces some stochastic orders.

Other stochastic order that could be defined is ordering distributions based on their arithmetic failure rates that we omitted it because it corresponds to the usual stochastic order. It is said that the non-negative random variable X with survival function \(\bar{F}_{X}\) is less (greater) than or equal with the non-negative random variable Y with survival function \(\bar{F}_{Y}\), and denote it by \(X\leq _{st}(\geq _{st})Y\) whenever \(\bar{F}_{X}(t)\leq (\geq )\bar{F}_{Y}(t)\), for all \(t\geq 0\). Note that \(X\leq _{st}Y\) if, and only if, \(E[\psi (X)]\leq E[\psi (Y)]\), for all increasing functions ψ for which the expectations exist (see Equation (1.A.7) in page 4 of Shaked and Shanthikumar [20]). To be more specific, \(A_{X}(t)=\frac{1}{t}\int _{0}^{t} h_{X}(x)dx\leq (\geq ) \frac{1}{t} \int _{0}^{t} h_{Y}(x)dx=A_{Y}(t)\), for all \(t\geq 0\) if, and only if, \(X(\leq _{st})\geq _{st}Y\). Some relations between the stochastic orders in Definition 2.1 with other well-known stochastic orders are brought in Bhattacharjee et al. [4]. From Shaked and Shanthikumar [20], it is said that X is less than or equal with Y in the hazard rate order if \(h_{X}(t)\geq h_{Y}(t)\), for all \(t\geq 0\). From Sengupta and Deshpande [19], it is said that X ages faster than Y and it is denoted by \(X\preceq _{c}Y\) when the ratio \(\frac{h_{X}(t)}{h_{Y}(t)}\) is increasing in \(t\geq 0\). Nanda et al. [15] showed that \(X\preceq _{c}Y\) implies that \(X\leq _{AI}Y\). Bhattacharjee et al. [4] proved that \(X\preceq _{c}Y\) yields \(X\leq _{GAI}Y\). They also showed that \(X\leq _{hr}Y\) implies \(X\leq _{GFR}Y\) and \(X\leq _{HFR}Y\). The following definition is also adopted from Roy and Mukherjee [18] and Bhattacharjee et al. [4].

It is said that X has an increasing (decreasing) failure rate (written as IFR (DFR)) when \(h_{X}(t)\) increases (decreases) in \(t>0\). From Lai and Xie [12], the random variable X is said to have increasing (resp. decreasing) failure rate on average (written as IFRA (resp. DFRA)) whenever \(A_{X}(t)\) is increasing (resp. decreasing) in \(t>0\). Nanda et al. [15] named this class increasing (resp. decreasing) aging intensity. Roy and Mukherjee [18] proved that IFR⊆IFRA⊆IGFR⊆IHFR and in addition DFR⊆DHFR⊆DGFR⊆DFRA.

As the failure rate may not be proportional over the whole time interval, but may be proportional differently in different intervals, Nanda et al. [16] proposed the dynamic proportional hazard rate (DPHR) model. Let X and \(X^{\star}\) be two non-negative random variables with hazard rate functions \(h_{X}\) and \(h_{X^{\ast}}\), such that where \(c(t)\) represents some non-negative function of t satisfying the following conditions (see Lemma 1.1 of Nanda et al. [16]): Nanda et al. [16] investigated the aging properties of X and \(X^{\star}\) in the DPHR model given in (5). They provided some conditions on \(c(t)\) under which some aging properties are transmitted from the random variables X into the random variable Y. They also presented sufficient conditions on \(c(t)\) under which X and \(X^{\star}\) satisfy some stochastic orders. It is notable that when \(h_{X}\) and \(h_{Y}\) are hazard rate functions, the function \(c(t)\) satisfies the conditions (i)-(iii). Bhattacharjee et al. [5] considered weighted distributions to produce the model (5) and called \(h_{Y}(x)\) the weighted failure rate.

In this paper, we also consider the dynamic PHR model in some specific cases as we explain here. A very well-known general family of distribution functions that generates the dynamic PHR model is the family of distorted distribution (or survival) functions. Let \(d:[0,1]\rightarrow [0,1]\) be a distortion function which must be increasing and right-continuous on \([0,1]\) such that \(d(0)=0\) and \(d(1)=1\) which means that \(d(\cdot )\) is a distribution function on \([0,1]\). Suppose first that \(\bar{F}_{X_{1}}\) is the survival function of the non-negative random variable \(X_{1}\) and consider then that the random variable \(X^{\ast}_{1}\) has distorted survival function \(\bar{F}_{X^{\ast}_{1}}(t)=d(\bar{F}_{X_{1}}(t)),~t\geq 0\). The distortion function \(d(\cdot )\) is assumed to be a differentiable function and we denote by \(d^{\prime}(\cdot )\) its derivative. Then, it is seen that \(X^{\ast}_{1}\) and \(X_{1}\) satisfy the dynamic PHR model given by \(h_{X_{1}^{\ast}}(t)=c(t)h_{X}(t)\) where \(c(t)= \frac{\bar{F}_{X_{1}}(t)d^{\prime}(\bar{F}_{X_{1}}(t))}{d(\bar{F}_{X_{1}}(t))}\). Next, we outline some specific cases of distorted distribution functions arisen out of a couple of physical situations. Let \(X_{1},\ldots ,X_{n}\) be n independent and identically distributed (i.i.d.) non-negative random variables with common survival function \(\bar{F}_{X_{1}}\). Then, the kth order statistic, \(X_{k:n}\), has a distorted survival function where \(d(u)=\sum _{j=n-k+1}^{n} \binom{n}{j}u^{j}(1-u)^{n-j}\). Let X and \(X_{k:n}\) have hazard rate functions \(h_{X_{1}}\) and \(h_{X_{k:n}}\). Then, it is seen that \(h_{X_{k:n}}(t)=\frac{h_{X_{1}}(t)}{\psi _{k}(\bar{F}_{X_{1}}(t))}\) where \(\psi _{k}(u)= \frac{\int _{0}^{u} y^{n-k} (1-y)^{k-1} dy}{(1-u)^{k-1} u^{n-k+1}}\), for \(0< u<1\) (see Misra and Francis [14]). The random variable \(X_{k:n}\) is the lifetime of an \((n-k+1)\)-out-of-n system which has been widely 1 applied in the context of reliability. Systems having a \((n-k+1)\)-out-of-n structure belong to a type of reliable coherent systems. A structure of order n that indicates that the entire system functions if, and only if, at least \(n-k+1\) components of the system function, where \(1\leq k \leq n\), is known as a \((n-k+1)\)-out-of-n structure. These structures are significant because, among all monotone structures of order n, the \((n-k+1)\)-out-of-n structure has the steepest dependability functions, as demonstrated by Barlow and Proschan [1]. Furthermore, because quantitative findings are simple to obtain. Therefore, these kinds of systems are highly beneficial from a mathematical perspective. The parallel and series systems that correspond to \(k=n\) and \(k=1\), respectively, are two significant special examples of \((n-k+1)\)-out-of-n systems. A common form of redundancy in fault-tolerant systems that has been extensively employed in both military and industrial systems is the \((n-k+1)\)-out-of-n system structure. The multi-display system in the cockpit, the multi-motor system in an airplane, and the multi-pump system in a hydraulic control system are examples of fault-tolerant systems.

There is another situation where the dynamic PHR model is arisen when \(X_{1},\ldots ,X_{n}\) are n dependent non-negative random variables with an Archimedean copula structure. Next, we bring some preliminaries concerning the notion of copula. Let \(\boldsymbol {X}=(X_{1},\ldots ,X_{n})\) be a vector of random variables with joint cdf F, survival function \({\overline{{\boldsymbol {F}}}}\) and marginal cdfs \(F_{i}\), \(i=1,\ldots ,n\). The function \(C :[0,1]^{n}\rightarrow [0,1]\) is said to be the copula function associated with the random vector X if If the cdf of \(X_{i}\), i.e. \(F_{i}\), is a continuous function, then the copula C is unique and it is defined as where \(F^{-1}\) denotes the quantile function of the random variable \(X_{i}\).

In a similar manner, a survival copula associated with a multivariate distribution function F is defined as The class of Archimedean copulas is a particularly interesting family of copulas. Archimedean copulas are widely used in reliability theory and actuarial mathematics because of their mathematical tractability and further because of their wide range of dependencies which can be entertained in practical situations. 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 pseudo-inverse of ϕ, a copula is called Archimedean if one can write where ϕ is called the generator of the Archimedean copula C. There are some assumptions regarding the generator function. The generator function ϕ has to fulfill that \((-1)^{k}\phi ^{[k]}(x)\geq 0\), for \(k = 0,\ldots , n-2\) such that \((-1)^{n-2}\phi ^{[n-2]}(x)\) is decreasing and convex, where \(\phi ^{[k]}(x)\) denotes the k-th derivative of the function \(\phi (x)\) with respect to x. Let \(X_{1},\ldots ,X_{n}\) be identically distributed dependent random variables with an Archimedean copula structure with generator function ϕ. Let \(X_{i}\)’s follow a common distribution function \(F_{X_{1}}\) and the associated survival function \(\bar{F}_{X_{1}}\). Let us denote by \(X_{n:n}^{\phi}\) the maximum order statistics of \(X_{1},\ldots ,X_{n}\). Then, it is seen that \(X_{n:n}^{\phi}\) has distribution function \(F_{X_{n:n}^{\phi}}(t)=\phi (n\phi ^{-1}(F_{X_{1}}(t))),~t\geq 0\). Therefore, \(h_{X_{n:n}^{\phi}}(t)=c(t)h_{X_{1}}(t)\) where \(c(t)=\gamma _{\phi}(F_{X_{1}}(t))\) in which \(\gamma _{\phi}(u)= \frac{n(1-u)\phi ^{\prime}(n\phi ^{-1}(u))}{(1-\phi (n\phi ^{-1}(u)))(\phi ^{\prime}(\phi ^{-1}(u)))}\) for all \(0< u<1\) (see Lemma 3.4). It is notable that \(X_{n:n}^{\phi}\) has a distorted survival function where \(d(u)=\phi (n\phi ^{-1}(1-u))\), where \(0< u<1\).

The aim of this paper is two issues. Firstly, we assume that X and \(X^{\star}\) satisfy the dynamic PHR model, i.e. . Then, we impose conditions on the function \(c(t)\) by which where represents one of the classes DHFR(IHFR), DGFR(IGFR), DHAI(IHAI) or the class DGAI(IGAI). Secondly, we study conditions for preservation of the underlying stochastic orders under the formation of the dynamic PHR model. Let us assume that \(X^{\ast}_{1}\) and \(Y^{\ast}_{2}\) have hazard rate functions \(h_{X^{\ast}_{1}}\) and \(h_{Y^{\ast}_{2}}\), such that \(h_{X^{\ast}_{1}}(t)=c_{1}(t)h_{X}(t)\) and \(h_{Y^{\ast}_{2}}(t)=c_{2}(t)h_{Y}(t)\). Then, we find conditions involving \(c_{1}(t)\) and \(c_{2}(t)\) under which where ≤_st−ord represents one of the stochastic orders \(\leq _{HFR},\leq _{GFR},\leq _{HAI}\) or ≤_GAI. As corollaries, we study whether the implications given in (6) and (7) hold true when \(X^{\ast}_{1}\) (or \(X^{\ast}\)) and \(Y^{\ast}_{2}\) are different order statistics arisen out of the samples \(X_{1},\ldots ,X_{n}\) and \(Y_{1},\ldots ,Y_{n}\), respectively, in independent case or dependent case with Archimedean copula as we outlined in the preceding paragraphs.

In this section, useful technical lemmas that will be used throughout the paper to obtain some corollaries of the main theorems are gathered. Before stating the main results we give definitions of a well-known stochastic order as we need it in the sequel. Let X and Y be two non-negative random variables with absolutely continuous distribution functions having density functions \(f_{X}\) and \(f_{Y}\), respectively. It is said that X is less than or equal with Y in the likelihood ratio order and denote it by \(X\leq _{LR}Y\) if \(\frac{f_{Y}(t)}{f_{X}(t)}\) is increasing in \(t>0\). From Theorem 1.C.1 of Shaked and Shanthikumar [20], \(X\leq _{LR}Y\) implies \(X\leq _{st}Y\). The bivariate non-negative function \(\phi (x,y)\) is said to be totally positive of order 2 (denoted by TP₂) on \(\mathfrak{X}\times \mathfrak{Y}\) if for all \(x_{1}\leq x_{2}\in \mathfrak{X}\) and for all \(y_{1}\leq y_{2} \in \mathfrak{Y}\), it holds that \(\phi (x_{1},y_{1})\phi (x_{2},y_{2})\geq \phi (x_{2},y_{1})\phi (x_{1},y_{2})\), where \(\mathfrak{X}\) and \(\mathfrak{Y}\) are two subsets of the real line. Let \(X_{1}\) and \(X_{2}\) be two non-negative random variables with absolutely continuous distribution functions having respective density functions \(f_{1}\) and \(f_{2}\), respectively. Then, \(X_{1}\leq _{LR}X_{2}\) if, and only if, \(f_{i}(x)\) is TP₂ in \(i=1,2\) and \(x>0\) (see, for instance, Joag-dev et al. [8]).

Below, a connection between the hazard rate of maximum order statistic of a dependent sample (with Archimedean copula) from an identical original distribution and the hazard rate of the original distribution is found.

In the following lemma, two important Archimedean copulas namely the Gumbel copula and the Clayton copula are considered.

In this section, we present some theorems giving the preservation properties of stochastic orders introduced in Sect. 2 under the dynamic PHR model given in (5). We also provide some corollaries of the main theorems when specific dynamic PHR models are considered. The following theorem presents sufficient conditions to get preservation of the harmonic failure rate order under dynamic PHR model.

The next corollary is resulted from Theorem 4.1 after using Lemma 3.1.

The following corollary is resulted from Theorem 4.1 and using Lemma 3.3(i) and Lemma 3.3(ii).

Next, we present a corollary of Theorem 4.1 where Lemma 3.4 and Lemma 3.5 are applied.

Notice that if in Corollary 4.4(i), \(\theta =1\) it means that the random variables \(X_{1},\ldots , X_{n}\) are i.i.d. and the random variables \(Y_{1},\ldots ,Y_{n}\) are also i.i.d. In this situation, the result of Corollary 4.3 where \(k=n\) and the result of Corollary 4.4 coincide with each other. Next, we give a theorem that establishes preservation of the geometric failure rate order under dynamic PHR model.

The corollary given below is concluded from Theorem 4.5 in which Lemma 3.1 applies. The proof is simple and hence we omit it.

The following corollary is resulted from Theorem 4.5 by applying Lemma 3.3(ii). The proof is easy and we omit it.

The next corollary is related to Theorem 4.5 when Lemma 3.4 and Lemma 3.5 are utilized. The proof is similar to that of Corollary 4.4 and hence we omit it.

Now, we obtain conditions under which the HAI order is preserved under dynamic PHR model.

The following corollary is result of application of Lemma 3.1 in Theorem 4.10.

Next, using Theorem 4.10 we impose conditions on two distortion functions under which one gets the harmonic aging intensity order in the dynamic PHR model.

The following corollary is resulted from Theorem 4.10 by applying Lemma 3.3(i) and Lemma 3.3(iii). The proof being straightforward is omitted.

The non-negative random variable \(X_{1}\) with cumulative distribution function \(F_{X_{1}}\) and probability density function \(f_{X_{1}}\), is said to have reversed hazard rate function \(\breve{h}_{X_{1}}(t)=\frac{f_{X_{1}}(t)}{F_{X_{1}}(t)},t>0\). The reversed hazard rate function of \(Y_{1}\), denoted by \(\breve{h}_{Y_{1}}\), is defined similarly. It is said that \(X_{1}\) is greater than \(Y_{1}\) in the reversed hazard rate order (denoted as \(X_{1}\geq _{rh}Y_{1}\)) provided that \(\breve{h}_{X_{1}}(t)\geq \breve{h}_{Y_{1}}(t)\), for all \(t>0\). The following corollary is deduced from Theorem 4.10 as Lemma 3.4, Lemma 3.5 and Lemma 3.6 are applied.

The following theorem imposes a condition to get the preservation of the GAI order under the dynamic PHR model.

The following corollary uses Theorem 4.15 and Lemma 3.1 to derive conditions under which the GAI order is preserved under the dynamic PHR model.

By utilizing Theorem 4.15 and applying Lemma 3.2 in it, we obtain conditions on two distortion functions such that the GAI order is satisfied in the dynamic PHR model.

We present a new corollary without proof that uses Theorem 4.15 Lemma 3.3(iii).

The corollary given below is a result of Theorem 4.15 where Lemma 3.4 and Lemma 3.6 apply. The proof is similar to that of Corollary 4.14 and hence we omit it.

In this section, we provide the results that present preservation properties of the new classes of lifetime distributions defined earlier in Sect. 2 under the structure of the dynamic PHR model. Below, we obtain a sufficient condition for preservation of the DHFR(IHFR) class.

Next, we derive a sufficient condition for preservation of the DGFR(IGFR) class.

The following corollary is deduced from Theorem 5.1 and Theorem 5.2 by using Lemma 3.1.

Below we give corollary of Theorem 5.1 and Theorem 5.2 in which Lemma 3.3(i) is used. This presents a useful application of preservation of the IHFR and IGFR classes under systems with \((n-k+1)\)-out-of-n structure. The reversed preservation property of the DHFR and DGFR classes under systems with such structure is also obtained.

Now, we provide a new corollary of Theorem 5.1 and Theorem 5.2 that is obtained by Lemma 3.4 and Lemma 3.5. This clarifies the preservation property of the IHFR and IGFR classes under parallel systems with dependent identical components. The reversed preservation properties of the DHFR and DGFR classes under such parallel systems are also presented.

Now, we find a condition to get the preservation of the DHAI(IHAI) class under the dynamic PHR model.

Next, we present a corollary of Theorem 5.6 by using Lemma 3.3(i) and Lemma 3.3(iii). The proof is very trivial and hence we omit it.

The following theorem examines preservation properties of DGAI and IGAI classes under dynamic proportional hazard rate model.

Below, we give corollary of Theorem 5.8 by means of Lemma 3.3(iii). The proof is obvious and hence we do not provide it.

A data set, presented in Table 1, of 29 time intervals between successive air conditioning failures in a Boeing 720 aircraft was reported by Proschan [17].

The gamma distribution with density function \(f(t)=\frac{\lambda ^{a}}{\Gamma (a)}t^{a-1}e^{-\lambda t}\), \(t\geq 0\), is considered for this data set and the maximum likelihood estimation of parameters is computed. The estimates are \(\hat{\lambda }=0.02\) and \(\hat{a}=1.67\). The p-value of the Kolmogorov–Smirnov, the Cramer–von Mises and the Anderson Darling goodness of fit tests are 0.9673, 0.8898 and 0.9268 which confirm that the gamma distribution could be a good describing model for this data set. The arithmetic and geometric mean failure rate functions are plotted in Fig. 5. The arithmetic and geometric mean failure rate functions of the time intervals related to a single air conditioning system is increasing. The plots show that as proved in this paper, the parallel system of two components and some considered k-out-of-n systems have increasing arithmetic and geometric mean failure rate functions. While the parallel system of two air conditioning systems shows a very smaller risk specially at early life stages. The 2-out-of-3 and 3-out-of-4 systems have better reliability at the beginning but the mean failure rate functions show higher risk for elderly system rather than one single air conditioning system.

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In this paper, newly proposed stochastic orders (initiated and defined by Bhattacharjee et al. [4]) which are constructed using harmonic hazard rate, harmonic aging intensity hazard rate, geometric hazard rate and geometric aging intensity hazard rate were applied on the dynamic proportional hazard rate model. In this model, we also investigated some relevant aging classes of lifetime distributions which has been defined by Roy and Mukherjee [18] and Bhattacharjee et al. [4]. Specifically, we studied whether these stochastic orders and related aging classes are transmitted from baseline (or parent) lifetime distribution into the response lifetime distribution which has a dynamic proportional hazard rate. Special examples in which the dynamic proportional hazard rate model is arisen have also been considered. These examples contain the general case of distorted distributions including two specific cases. The lifetime of a coherent system with \((n-k+1)\)-out-of-n structure where the components have an identical distribution with independent components follows a distorted distribution. The lifetime distribution of this system corresponds to a response lifetime distribution with dynamic proportional hazard rate so that the common lifetime distribution of the components plays the role of the baseline distribution. The maximum order statistic of a sample of non-negative dependent random variables which have an identical distribution and they are connected via the Archimedean copula also considered as a lifetime random variable with a response lifetime distribution in the dynamic proportional hazard rate.