Stochastic comparisons of series and parallel systems with dependent and heterogeneous Topp-Leone generated components

: In this paper, we carry out stochastic comparisons of lifetimes of series and parallel systems with dependent heterogeneous Topp-Leone generated components. The usual stochastic order and the reversed hazard rate order are developed for the parallel systems with the help of vector majorization under Archimedean copula dependence, and the results for the usual stochastic order are also obtained for the series systems. Finally, some numerical examples are provided to illustrate the e ﬀ ectiveness of our theoretical ﬁndings

comparing systems with independent heterogeneous components under specific distributions.For more on this topic, please see Dykstra et al. [12], Balakrishnan et al. [13] and Torrado [14].However, the components of the system share many complex factors when the system is functioning, such as environmental conditions and work stress.For these reasons, it would be practicable to consider dependent lifetimes of components.Recently, the dependence structure of the components is investigated by researchers with the help of copula theory.Archimedean copula has been considered by many scholars due to its flexibility, for instance, Clayton copula, Ali-Mikhail-Haq copula, and Gumbel-Hougaard copula.For example, Navarro and Spizzichino [15] studied the stochastic ordering of series and parallel systems with components sharing a common copula.Li and Li [16] considered ordering properties of the smallest order statistics of Weibull samples having a common Archimedean copula.Li et al. [17] discussed stochastic comparisons of extreme order statistics from scaled and interdependent random variables.Fang et al. [18] presented conditions to stochastically compare the extreme order statistics from dependent and heterogeneous random variables.
Kundu and Chowdhury [19] discussed the lifetimes of two series and parallel systems with location-scale components assembled with some kind of Archimedean copulas under different stochastic orders.Fang et al. [20] obtained various ordering results for comparing the lifetimes of the series and the parallel systems, where each component follows scale proportional hazard or reversed hazard models with Archimedean copula.For proportional hazard rate and proportional reversed hazard rate models, Li and Li [21] developed ordering properties of extreme order statistics from heterogeneous dependent random variables in the sense of the hazard rate and the reversed hazard rate orders.
In this manuscript, we study the stochastic comparisons of series and parallel systems having Topp-Leone generated components with different scale and shape parameters.The Topp-Leone generated family of distribution was given by Sadegh Rezaei [22] as a generalization of Topp and Leone's distribution.It has the property to model bathtub-shaped hazard rates depending on the values of parameters and can be used for lifetime modeling.For more applications of the distribution, one may refer to Sadegh Rezaei [22].A random variable X is said to be Topp-Leone generated (T L − G) family of distribution if its cumulative distribution function is where θ is the scale parameter, α is the shape parameter, G(x; ξ) is the baseline distribution function, and ξ is the parameter specifying the baseline distribution, and denote X ∼ T L − G(α, θ, ξ).For convenience, G(x; ξ) is written by G(x).T L − G(α, θ, ξ) is reduced to Topp and Leone's distribution when G(x) ∼ U(0, 1) and θ = 1.
The organization of the paper is as follows.In Section 2, we present some fundamental definitions and lemmas.In Section 3, we develop the usual stochastic and the reversed hazard rate orders of series and parallel systems with dependent heterogeneous components under Archimedean copulas.Some numerical examples are provided to illustrate theoretical findings.Section 4 concludes the paper.

Preliminaries
In this section, we first recall some basic definitions of some well-known notions of stochastic orders, majorization orders, and Archimedean copula, and introduce some lemmas may be used in the sequel.Denote R = (−∞, +∞) and R n = (−∞, +∞) n .
X ≤ st Y means X is less likely than Y to take on large values, where "large" means any value greater than x, and that this is the case for all x s.The reversed hazard rate could be understood as the probability intensity of a component survival to the last moment t given that its lifetime does not exceed t (cf.Yan and Luo [23]).The hazard rate is well known and has been widely applied.As a dual concept of the hazard rate, the reversed hazard rate is far less popular and frequently used.Recently, the reversed hazard rate order has received great attention because it is more appropriate and effective than the hazard rate order for the study of some particular problems such as assessing waiting time, hidden failures, inactivity times, etc. (cf.Veres-Ferrer and Pavia [24]).
It is well known that the hazard rate order or the reversed hazard rate order implies the usual stochastic order, and not conversely.For more detailed discussions and applications of stochastic orders, please refer to Shaked and Shanthikumar [25].In addition, some scholars recently have compared order statistics under some weaker orders such as the second-order stochastic dominance, the details can be referred to Lando et al. [26,27].
Let x 1:n ≤ x 2:n ≤ • • • ≤ x n:n and y 1:n ≤ y 2:n ≤ • • • ≤ y n:n be the increasing arrangements of the elements of vectors x = (x 1 , . . ., x n ) and y = (y 1 , . . ., y n ), respectively.In particular, x ≤ y means x i ≤ y i , for all i = 1, . . ., n. Definition 2. For two vectors x = (x 1 , . . ., x n ) and y = (y 1 , . . ., y n ), x is said to (i) majorize y (written as x m y) if j i=1 x i:n ≤ j i=1 y i:n , j = 1, . . ., n − 1, and n i=1 x i:n = n i=1 y i:n ; (ii) supermajorize y (written as x w y) if j i=1 x i:n ≤ j i=1 y i:n , j = 1, . . ., n; (iii) submajorize y (written as x w y) if n j=i x j:n ≥ n j=i y j:n , i = 1, . . ., n.The majorization, the weak supermajorization order, and the weak submajorization order are widely used to establish various stochastic inequalities.For more details on the notions and basic properties of majorization orders, one may refer to Marshall [28].
The following we recall the notions of copula.
Definition 3.For a random vector X = (X 1 , X 2 , . . ., X n ) with joint distribution function F, joint survival function F, univariate distribution functions F 1 , F 2 , . . ., F n and univariate survival functions F1 , . . ., Fn .If there exist some function C : then C and C are called the copula and survival copula of random vector X, respectively.
For more on Archimedean copula, readers may refer to Nelsen [29].For convenience, from now on, we denote Definition 5. A function f is said to be super-additive if f (x + y) ≥ f (x) + f (y), for all x and y in the domain of f .Definition 6.A real-valued function ϕ defined on a set A ⊆ R n is said to be Schur-convex In the following, we present some lemmas will be used in proving the main results.
Lemma 1. (Li and Fang [30].Lemma A.1) For two n-dimensional Archimedean copulas C Lemma 2. (Marshall et al. [28].Theorem 3.A.4) Suppose J ⊂ R is an open interval and φ : J n → R is continuously differentiable.Then, φ is Schur-convex [Schur-concave] on J n if and only if (i) φ is symmetric on J n ; and (ii) for all i j and all x ∈ J n , where ∂φ(x) ∂x i represents the partial derivative of φ with respect to its i-th argument.Lemma 3. (Marshall et al. [28].Theorem 3.A.8)A real valued function φ on R n , satisfies if and only if φ is increasing and Schur-convex [Schur-concave] on R n .Lemma 4. (Das and Kayal [31].Lemma 2.6) Let the function h : (0, 1) → (−∞, 0) be defined as h(u) = u ln u 1−u .Then, h(u) is decreasing in u for all u ∈ (0, 1).
Lemma 5. (Li and Li [21].Lemma 2.3) If λ w µ, ψ is log-concave and (ψ ln ψ)/ψ is increasing and concave, then Throughout this paper, the terms increasing and decreasing are used for non-decreasing and nonincreasing, respectively.It is also assumed that all the random variables are non-negative and absolutely continuous.

Main results
In this section, we compare two system lifetimes with dependent heterogeneous components following the Topp-Leone generated distribution.The usual stochastic order and the reversed hazard rate order are obtained for series and parallel systems.Let X = (X 1 , . . ., X n ) be the dependent heterogeneous TL-G distributed random vectors, we denote X ∼ TL-G(α, θ, F), where F is the baseline distribution function, ψ 1 is generator of the associated Archimedean copula, and α = (α 1 , . . ., α n ) and θ = (θ 1 , . . ., θ n ) are the tilt parameter vector.
Similarly, denote Y ∼ TL-G(β, δ, G).For convenience, we denote the distribution(survival) function of random variable X by H X (x)( HX (x)).Then, the distribution functions of the parallel systems X n:n and Y n:n are given by respectively.And the reliability functions of series systems X 1:n and Y 1:n can be written as respectively, where φ 1 = ψ −1 1 and φ 2 = ψ −1 2 .First, we develop sufficient conditions for the usual stochastic order between parallel systems with dependent components.In the following theorem, we consider the case of the heterogeneous shape parameters and common scale parameters.Theorem 1. Suppose that X and Y are two random variables with distribution functions F and G, respectively.Define X ∼ TL-G(α, θ, F) with generator ψ 1 and Y ∼ TL-G(β, θ, G) with generator ψ 2 , such that α, β and θ ∈ + .If X ≥ st Y, φ 2 • ψ 1 is super-additive, and either ψ 1 or ψ 2 is log-convex.Then, Proof.Without loss of generality, assume that ψ 2 is log-convex, it follows from Lemma 1 that By the decreasing property of ψ 2 and φ 2 , and note that X ≥ st Y, we have Combining (3.1) and (3.2), we conclude Thus, to prove the required result, it is sufficient to show that On differentiating (3.3) with respect to α i (i = 1, 2, . . ., n), we obtain Hence, ϕ 2 (α, θ, G, ψ 2 ) is decreasing in α i (i = 1, 2, . . ., n).After simplifications, we have Under the assumption of α, θ ∈ + , we have From the log-convexity of ψ 2 that holds It follows immediately from Lemma 2 that ϕ 2 (α, θ, G, ψ 2 ) is Schur-concave in α.Note that α w β, we obtain that ϕ 2 (α, θ, G, ψ 2 ) ≤ ϕ 2 (β, θ, G, ψ 2 ) by Lemma 3. The proof is completed.
Remark 1. (i) Note that the super-additivity of φ 2 •ψ 1 can be roughly interpreted as follows: Kendall's τ of the copula with generator ψ 2 is larger than that with generator ψ 1 and hence is more positive dependent (refer to Li and Fang [30]).Theorem 1 shows that the less positive dependence, and the more heterogeneous shape parameters (in the weakly supermajorized order) lead to the stochastically large lifetime of the parallel system and higher reliability could be achieved.
(iii) To display the whole curves of distribution functions of random variables defined on [0, +∞), we take the transformation (x + 1) Through this transformation, we can display the graphs of the CDFs in the (0, 1] interval and further realize the panoramic observation of random variables.
The following example 1 illustrates the theoretical result of Theorem 1.
Remark 2. In accordance with Theorem 2, for a series system, the more positive dependence, and the less heterogeneous (in the weakly subermajorized order) the shape parameters are, the stochastically longer lifetime of the system and hence higher reliability will be achieved.It should be pointed out that Theorem 2 expands Theorem 3.5 of Chanchal et al. [32] to the case of the heterogeneous components.
It might be of great interest to study the hazard rate order between X 1:n and Y 1:n , which is left as an open problem.
The following example 2 illustrates that the theoretical result of Theorem 2.
Then, the conditions of Theorem 2 are satisfied.Figure 2 plots distribution functions of (X 1:2 + 1) −1 and (Y 1:2 + 1) −1 , from which it can be observed that H (X 1:2 +1) −1 (x) is always smaller than H (Y 1:2 +1) −1 (x), and this verifies X 1:2 ≤ st Y 1:2 .The next theorem establishes some sufficient conditions for the usual stochastic order, here, we consider different scale parameter vectors.Theorem 3. Suppose that X and Y are two random variables with distribution functions F and G, respectively.Define X ∼ TL-G(α, θ, F) with generator ψ 1 and Y ∼ TL-G(α, δ, G) with generator ψ 2 , such that θ, δ and α Proof.First assume that ψ 2 is log-convex.Similar to the arguments as in the proof of Theorem 1, we notice that to prove the present theorem, it is enough to show that On differentiating (3.9) with respect to θ i (i = 1, 2, . . ., n), we have It follows that ϕ 2 (α, θ, G, ψ 2 ) is decreasing in θ i (i = 1, 2, . . ., n).We further get Further, we have By the log-convexity of ψ 2 , we conclude For convenience, denote Then, Eq (3.9) is equivalent to ) by Lemma 3. The desired result is proved.Remark 3. Theorem 3 manifests that the less positive dependence, and the more heterogeneous scale parameters (in the weakly supermajorized order) lead to the stochastically large lifetime of the parallel system and higher reliability could be achieved.It should be noted that Theorem 3 generalizes Theorem 3.2 in Chanchal et al. [32] to the case of the dependent components.It might be of great interest to establish the (reversed) hazard rate order between X n:n and Y n:n of Theorem 3, which is left as an open problem.
The following example 3 illustrates the theoretical result of Theorem 3.
Proof.Similar to the proof of Theorem 2, we only need to prove that (3.10) On differentiating (3.10) with respect to θ i (i = 1, . . ., n) give rises to , hence the proof is finished.
Remark 4. It should be mentioned that the condition φ 2 • ψ 1 (x) is super-additive in Theorem 4 is quite general and easy to be constructed for many well-known Archimedean copulas.For example, consider Ali-Mikhail-Haq(AMH) copula with generator ψ(x) = 1−a e x −a for a ∈ [0, 1), it is easy to see that ln It is natural to ask whether the condition in Theorem 4 can be weakened ?The following numerical example provides a negative answer.We show that if we take θ m δ , then the result in Theorem 4 does not hold.
Remark 5. Theorem 5 states that less heterogeneous shape parameters in the weakly supermajorized order lead to a large lifetime of the parallel system in the sense of the reversed hazard rate order.It might be of great interest to study the likelihood ratio order between X n:n and Y n:n , which is left as an open problem.
The following example 4 illustrates that the theoretical results of Theorem 5.

Conclusion
In this paper, we study stochastic comparisons of series and parallel systems with heterogeneous Topp-Leone generated components with Archimedean (survival) copulas.We established the usual stochastic order of the series and parallel systems, and the reversed hazard rate order of the parallel system.These results generalize some existing results in the literature (see Chanchal et al. [32]) to the case of dependent components in the T L-G family.