Stochastic comparisons of second-order statistics from dependent and heterogeneous modiﬁed proportional (reversed) hazard rates scale models

: In this paper, we investigate the problem of stochastically comparing the second-order statistics from dependent and heterogeneous samples following modiﬁed proportional hazard rates scale (MPHRS) and modiﬁed proportional reversed hazard rates scale (MPRHRS) models under Archimedean copula. We built some su ﬃ cient conditions for the usual stochastic order whenever the samples have di ﬀ erent parameter vectors. Finally, some numerical examples were provided to illustrate the theoretical results.


Introduction
Order statistics play an important role in reliability theory, auction theory, operations research, and many applied probability areas.X k:n denotes the kth smallest of random variables X 1 , . . ., X n , k = 1, . . ., n.In reliability theory, X k:n characterizes the lifetime of a (n − k + 1)-out-of-n system, which works if at least n − k + 1 of all the n components function normally.Specifically, X 1:n and X n:n denote the lifetimes of series and parallel systems, respectively.In auction theory, X 1:n and X n:n represent the final price of the first-price procurement auction and the first-price sealed-bid auction (see [1]), respectively.G. Pledger et al. [2] was the first to deal with the problem of comparing order statistics from heterogeneous exponential random variables.Subsequently, many researchers devoted themselves to stochastic comparisons of order statistics from heterogeneous independent or dependent samples; to name a few, see [3][4][5][6][7][8][9][10].
In this paper, we focus on second-order statistics, which also have a very wide application background.In auction theory, X 2:n denotes the winner's price for the bid in the second-price reverse auction [11].In the reliability context, the second-order statistic X 2:n characterizes the lifetime of the (n−1)-out-of-n system in reliability theory (referred to as the fail-safe system; see [12]).E. Pǎltǎne [13] established the hazard rate order for comparing second-order statistics from heterogeneous exponential random variables.P. Zhao et al. [14] further extended the result of [13] from the hazard rate order to the likelihood ratio order.P. Zhao et al. [15] examined the mean residual life order between the secondorder statistics from two sets of exponential random variables.P. Zhao et al. [16] studied the stochastic comparison of fail-safe systems with heterogeneous exponential components in terms of the dispersive order.N. Balakrishnan et al. [17] investigated the stochastic comparison of the second-order statistics from independent heterogeneous and homogeneous samples having different sample sizes in the sense of mean residual life, dispersive, hazard rate, and likelihood ratio orderings.X. Cai et al. [18] compared the hazard rate functions of the second-order statistics arising from two sets of independent multipleoutlier proportional hazard rates samples.For dependent and heterogeneous samples, R. Fang et al. [19] conducted stochastic comparisons on sample minimums (maximums) and the second smallest (largest) order statistic from proportional hazard rate and the proportional reversed hazard rates models.Additionly, C. Li et al. [20] obtained the usual stochastic order of the sample extremes and the second smallest order statistic from the scale model.T. Lando et al. [21] dealed with the increasing concave comparison of k-order statistics (iid case) for wide nonparametric families.S. Das et al. [22] considered stochastic comparisons between second-order statistics arising from general exponentiated locationscale models when the random variables are independent, and established usual stochastic and hazard rate orders between second-order statistics.O. Shojaee et al. [23] provided sufficient conditions to compare the smallest and the second smallest (largest and second largest) order statistics of dependent and heterogeneous random variables having the additive hazard model with the Archimedean copula in the sense of usual stochastic order and hazard rate order.R. F. Yan [24] studied the stochastic comparisons of the second-order statistics from dependent or independent and heterogeneous modified proportional hazard rate observations.G. Barmalzan et al. [25] studied the second smallest and the second largest order statistics from a general semiparametric family of distributions.
In reliability theory, to model the lifetime data with different hazard shapes, it is desirable to introduce flexible families of distributions.To this end, A. W. Marshall et al. [26] developed a new method to introduce one parameter to a base distribution, resulting in a new family of distribution with more flexibility.For a baseline distribution function F with support R + = (0, ∞) and corresponding survival function F, for any and are two new defined distribution functions, where the parameter α is called a tilt parameter ( ᾱ = 1 − α) (see [27]).Note that (1.1) is equivalent to (1.2) if α in (1.1) is changed to 1/α.The proportional hazard rates (PHR) and the proportional reversed hazard rates (PRHR) models have important applications in reliability engineering and operations research ( [28][29][30][31][32][33]). The random variables is the resilience vector.It is well-known that the exponential, Weibull, Lomax, and Pareto distributions are special cases of the PHR model, and the Fréchet distribution is a special case of the PRHR model.N. Balakrishnan et al. [34] introduced the modified proportional hazard rates (MPHR) and modified proportional reversed hazard rates (MPRHR) models by adding a parameter to the PHR and PRHR models, respectively.For any α, λ, β ∈ R + , their respective distributions are given by where λ and β are the PHR and PRHR parameters, respectively.They established some stochastic comparison results between the corresponding order statistics with independent samples.G. Barmalzan et al. [35] discussed the hazard rate order and reversed hazard rate order of series and parallel systems with dependent components following either MPHR or MPRHR models under Archimedean copula.M. M. Zhang et al. [36] investigated stochastic comparisons on extreme order statistics from dependent and heterogeneous samples following MPHR and MPRHR models, and built the usual stochastic order for sample minimums and maximums, the hazard rate order on minimums of sample and the reversed hazard rate order on maximums of sample.S. Das et al. [37] introduced a scale parameter into the MPHR and MPRHR model that leads to new models, which are called modified proportional hazard rate scale (MPHRS) and modified proportional reversed hazard rate scale (MPRHRS) models.For any α, λ, θ, β ∈ R + , are two newly defined distribution functions, respectively.They also obtained some stochastic comparison results on independent samples in terms of the usual stochastic, (reversed) hazard rate orders.The MPHRS (MPRHRS) model contains the MPHR (MPRHR), PHR (PRH), and scale models as special cases, and the flexibility that it possesses makes it quite suitable for modeling reliability and data analysis.Motivated by the work of [19,20], in this paper, we consider two samples from dependent and heterogeneous MPHRS and MPRHRS models.Some sufficient conditions are established to stochastically compare the second smallest (largest) order statistics of two samples with different parameters in the sense of the usual stochastic order.
The remainder of this paper is organized as follows: Section 2 recalls some concepts and notations used in this paper.Section 3 presents the main results and provides some sufficient conditions under which the two samples from the MPHRS model are stochastically comparable in the sense of the usual stochastic order.Also, similar results are obtained for the case of samples following the MPRHRS model in terms of the usual stochastic order.Section 4 presents the applications of the obtained results.Section 5 concludes the paper.

Preliminaries
In this section, let us first recall some important concepts and notations related to the main results of this article.
For random variables X and Y, let F and G be distribution functions ( f and g be densities when absolutely continuous), and denote F = 1 − F and Ḡ = 1 − G as their reliability functions and h and r as their hazard rate and reversed hazard rate functions, respectively.Denote I n = {1, . . ., n} and Definition 1.For two nonnegative random variables X and Y, X is said to be smaller than Y in the For more comprehensive discussions on stochastic orders, please refer to [38,39].Next, we introduce the notions of majorization and related orders, which are key tools in establishing various inequalities arising from many research areas.For two real vectors x = (x 1 , ..., x n ) and y = (y 1 , ..., y n ) ∈ R n , denote the increasing arrangement of the components of x and y by Definition 2. The vector x is said to be (i) majorized by the vector y (write as , for all j ∈ I n−1 , and n i=1 x (i) = n i=1 y (i) ; (ii) weakly supermajorized by the vector y (write as x w y ) if j i=1 x (i) ≥ j i=1 y (i) , for all j ∈ I n ; (iii) p-large than the vector y (write as x p y ) if n i= j x (i) ≥ j i=1 y (i) , for all j ∈ I n .
Definition 3. A real function defined on A ⊆ R n is said to be Schur-convex (Schur-concave) on A if It is well-known that x m y ⇒ x w y ⇒ x p y for any x, y ∈ R n , while the converse is not always true.For more details on majorization and Schur-convexity (Schur-concavity), please refer to [40].Now, let us review the concept of Archimedean copulas.
For detailed discussions on copulas and their applications, please refer to [41].The following lemmas are useful to establish the main results.Lemma 1. [42] Let I ⊆ R be an open interval.A continuously differentiable : I n → R is Schurconvex (Schur-concave) if and only if is symmetric on I n , and for all i j [40] For a real function on A ⊆ R n , x w y implies (x) ≤ (≥) (y) if and only if is decreasing (increasing) and Schur-convex (Schur-concave) on A.
Throughout the manuscript, all concerned random variables are assumed to be absolutely continuous and nonnegative, and the terms increasing and decreasing stand for nondecreasing and nonincreasing, respectively.

Results
In this section, we present the stochastic comparison results of the second smallest (largest) order statistics from dependent and heterogeneous MPHRS (MPRHRS) samples in the sense of the usual stochastic order.
The first result presents the result for comparing the samples with different tilt parameters in the sense of the usual stochastic order.Theorem 1.For X ∼ MPHRS (α; θ1; λ1; F1 ; ψ) and Y ∼ MPHRS (β; θ1; λ1; Proof.The survival function of X 2:n can be written as It is easy to obtain that It is needed to show that J 2 α, θ, λ, ψ, F2 (x) is increasing in α k , k ∈ I n and Schur-concave in α.
By the decreasing and convex property of ψ, for α k ≥ (≤)α l , we have It is easy to verify that 1/∆ 2 (α, x) is negative and increasing in α for the given x ≥ 0. Thus, for k l, That is, X 2:n ≥ st Y 2:n .The proof is completed.The next corollary follows immediately from Theorem 1.
The next theorem investigates the impact of the scale vector on the second smallest samples with respect to the usual stochastic order, whenever other parameters are equal.
Remark 1. Theorem 4.3 of [20] developed a similar result for the scale model, which can be treated as a special case of the above result when α = λ = 1 and F1 = F2 .As a further study, it is of interest to obtain the results for the case α > 1.
As a consequence of Theorem 2, we obtain the following corollary immediately.
First, we establish sufficient conditions for the usual stochastic order when the samples have different modified proportional reversed hazard rate vectors.
Theorem 3.For X ∼ MPRHRS (α1; θ1; λ; F 1 ψ) and Y ∼ MPRHRS (α1; θ1; µ; F 2 ; ψ), where Proof.The distribution function of X n−1:n and Y n−1:n can be expressed as We need to show that It is easy to obtain that Note that ψ is decreasing and convex.It holds that ψ is increasing and nonpositive.As a result, differentiating Q 1 α, θ, λ, ψ, F 2 (x) with respect to λ k , we have where .
By the decreasing and convex property of ψ, for λ k ≥ (≤)λ l , it holds that It is easy to check that Λ 1 (λ, x) is nonpositive and increasing in λ, then, for k l, , then, we have X n−1:n ≥ st Y n−1:n , which completes the proof.
Remark 2. Theorem 6.2 of [19] obtained a similar result for the PRH model, which can be regarded as a special case of the above result when α = θ = 1 and F 1 = F 2 .
Proof.The distribution function of X n−1:n can be written as It is easy to obtain that It is needed to show that Q 2 α, θ, λ, ψ, F 2 (x) with respect to a k is increasing and Schur-concave in α, The rest of this part can be proven in a similar manner with Theorem 1 and is thus omitted.The next corollary follows immediately from Theorem 4.
One natural question is whether the condition of weakly supermajorization order can be replaced by p-large order in Theorem 4. The following example gives a negative answer.3, from which we confirm that X 2:3 st Y 2:3 and X 2:3 st Y 2:3 .

Reliability theory
In reliability theory, the k-out-of-n system as the popular fault tolerant system has been widely applied in industrial engineering and system security.Specifically, X 1:n and X n:n denote the lifetimes of series and parallel systems, respectively, and the (n − 1)-out-of-n system is referred to as the fail-safe system.
Consider a fail-safe system with dependent and heterogeneous components lifetime following the MPHRS model.Theorem 2 states that the larger the baseline survival function for the negatively lower orthant dependent components, the larger heterogeneity among the scale parameters, which leads to the better performance of the fail-safe system, respectively.Theorem 1 states that the larger the baseline survival function and the more symmetric the tilt parameter vector leads to a more reliable fail-safe system.

Auction theory
The second-price sealed-bid auction is of important theoretical and practical interest in auction theory.Several bidders compete to buy a good, and bidders hand in their bids to the auctioneers simultaneously without the knowledge of their rivals' bids.The bidder with the highest bid wins the object and pays the second highest bid in the English auction.While in a second-price reversed auction, the lowest bidder wins and is paid at a price corresponding to the second lowest bid.The winner will pay the rent defined as the difference between his or her bid and the final price for the auctioneer.
In a second-price reversed auction, there are several bidders with dependent and different bids following the MPHRS model with negatively lower orthant dependence.The larger the baseline survival function, the more symmetric scale and modified proportional hazard rate vectors lead to stochastically larger the revenue of the auctioneer, and the more symmetric tilt parameter vector incurs to stochastically lower the revenue of the auctioneer.In the second price auction for bids following the MPRHRS model, Theorem 3 (Theorem 4) states that the more symmetric modified proportional hazard rate vector (tilt parameter vector) with a lower (higher) baseline distribution function will lead to stochastically lower (higher) the revenue of the auctioneer.

Conclusions
In this paper, we study the problem of stochastically comparing the second smallest (largest) order statistics from dependent and heterogeneous samples.For the second smallest order statistics from MPHRS samples, sufficient conditions are obtained for the usual stochastic order whenever the sample has different parameters.In addition, similar results are established for the second largest order statistics from MPHRS samples.Lastly, some applications of the obtained results in reliability theory and auction theory are provided.

Use of AI tools declaration
The author declares he has not used Artificial Intelligence (AI) tools in the creation of this article.