Dispersive order comparisons on extreme order statistics from homogeneous dependent random vectors

In this paper, we investigate suﬃcient conditions for preservation property of the dispersive order for the smallest and largest order statistics of homogeneous dependent random vectors. Moreover, we establish suﬃcient conditions for ordering with the dispersive order the largest order statistics from dependent homogeneous samples of diﬀerent sizes.


Introduction
Order statistics play an important role in statistics, risk management, auction theory, reliability and many other theoretical and applied probability areas.They have received a lot of attention from many researchers.For comprehensive references one may refer to Balakrishnan and Rao ( [1], [2]) and David and Nagaraja [3].
For a random vector X = (X 1 , . . ., X n ), denote as X i:n the corresponding ith order statistic, i = 1, . . ., n.Most of the research on stochastic comparisons between order statistics has been dedicated to the case of independent and identically distributed (i.i.d.) random variables X 1 , . . ., X n .We can quote Boland et al. [4] who prove stochastic comparisons between order statistics with the hazard rate order and the likelihood ratio order, Raqab and Amin [5] who further prove comparison with the likelihood ratio order between order statistics from samples of different sizes, Kochar [6] who proves comparisons with the dispersive order between order statistics for decreasing failure rate (DFR) distributions or Khaledi and Kochar [7] who study comparisons with the dispersive order between order statistics from samples of different sizes for DFR distributions.
In practical situations, the observations are usually not i.i.d.During the last three decades, the case of independent but not necessarily identically distributed random variables has also got the attention of researchers.We refer the reader to Kochar [8] and Balakrishnan and Zhao [9] for comprehensive references.To mention a few with the dispersive order, Dykstra et al. [10] study comparisons of the largest order statistics with the dispersive order for independent exponential random variables.Khaledi and Kochar [11] extend the latter result from the exponential case to the proportional hazard PH) sample.More recently, Fang and Zhang [12] have obtained the dispersive order between maximums of one heterogeneousand one homogeneous-independent samples for Weibull random variables sharing a common shape parameter.
In the last decade, some papers have been devoted to the study of the ordering properties of the order statistics from dependent samples.For example, Li and Fang [13] generalize the result in Fang and Zhang [12] to the dependent case where two proportional hazards samples have a common Archimedean copula and one has heterogeneous hazards and the other has the homogeneous arithmetic average hazards.For two Weibull samples having a common Archimedean survival copula, Li and Li [14] further prove dispersive order inequalities between minimums of one heterogeneous and one homogeneous samples.Fang et al. [15] derive the usual stochastic order, the dispersive order and the star order of order statistics from the PH sample with Archimedean survival copulas and the proportional reversed hazards (PRH) sample with Archimedean copulas.Li et al. [16] investigate order statistics from random variables following the scale model and obtain in the presence of the Archimedean copula or survival copula the usual stochastic order of the sample extremes and the second smallest order statistic, the dispersive order and the star order of the sample extremes.
In this paper, we consider homogeneous dependent samples and we establish sufficient conditions on the copulas or survival copulas and the marginal distribution functions in order to preserve the dispersive order for the smallest and largest order statistics.Moreover, we obtain sufficient conditions on the copulas and the marginal distribution functions for ordering with the dispersive order the largest order statistics from dependent homogeneous samples of different sizes.
The paper is structured as follows.In Section 2, we introduce the useful concepts that will be used in the rest of the paper.Section 3 is devoted to homogeneous dependent samples with different copulas and marginal distributions, and we obtain sufficient conditions for preserving the dispersive order for the smallest and largest order statistics.Finally, in Section 4, we derive sufficient conditions for the dispersive order of the sample extremes in the case of dependent homogeneous samples of different sizes with common copulas and marginal distributions.

Preliminaries
In this section, we recall the concepts that are important in the following.
Let X and Y be two random variables with their respective distribution functions F and G, survival functions F and Ḡ and right continuous inverses F −1 and G −1 .Definition 2.1.X is said to be smaller than Y in the dispersive order, denoted For more details on stochastic orders, we refer the reader to Shaked and Shanthikumar [17].
Let X = (X 1 , . . ., X n ) be a random vector with joint distribution function F , joint survival function F , univariate marginal distribution functions F 1 , . . ., F n , survival functions F1 , . . ., Fn and right continuous inverses for all (u 1 , . . ., u n ) ∈ [0, 1] n , then C and Ĉ are called the copula and the survival copula of X, respectively.The functions δ C (u) = C(u, . . ., u) and δ Ĉ (u) = C(u, . . ., u) are known as the diagonal sections of C and Ĉ.The Archimedean copulas form an important class of copulas.
where we denote ψ = φ −1 the right continuous inverse of φ for convenience.

Comparison of the smallest and largest order statistics with the dispersive order
Let X = (X 1 , . . ., X n ) and Y = (Y 1 , . . ., Y n ) be two homogeneous random vectors with copulas C X and C Y and univariate marginal distribution functions F and G, respectively.Under the conditions that both C X and C Y are the independence copula, Theorem 3.B.26 in Shaked and Shanthikumar [17] states that if X disp Y , then X i:n disp Y i:n for i = 1, . . ., n, where X i:n and Y i:n are the ith order statistics of X and Y , respectively.As mentioned in Shaked and Shanthikumar [17], this preservation property of the dispersive order is useful in reliability theory and in nonparametric statistics.Actually, the latter result can be extended for X and Y having a common copula, as shown next.
Proof.Since X and Y have a common copula, their exists a uniform random vector (U 1 , . . ., U n ) with distribution C X such that This ensures that the distributions of X i:n and Y i:n are of the form , where H i:n denotes the distribution of U i:n .The announced result then follows from Proposition 3.1 considers samples X and Y with a common copula.The next result show that the dispersive order can still be preserved for the smallest and largest order statistics when we consider different copulas for X and Y at the cost of requiring additional conditions on the marginal distribution G and the diagonal sections δ C X and δ C Y of the copulas C X and C Y for the largest order statistics, and on the survival function Ḡ and the diagonal sections δ ĈX and δ ĈY of the survival copulas ĈX and ĈY for the smallest order statistics.
Proof.(i) Obviously, the quantile functions of X n:n and Y n:n are } is decreasing in α.This latter difference can be rewritten as the sum of ) is an increasing function), it suffices to prove that B(α) is decreasing in α as well.Simple calculations yield where Finally, since G is log-convex, it is plain that h is a decreasing function so that the right-hand side in (3.2) is negative.
(ii) The quantile function of X 1:n is clearly given by F −1 Proceeding in a similar manner than in (3.1) and using the fact that δ −1 where h(t) = Ḡ(t)/ Ḡ (t).Consequently, as Ḡ is log-convex, it is plain that h is a decreasing function so that the right-hand side in (3.3) is positive.
In particular, for Archimedean copulas, we directly get the following result.
with a ∈ R can be expressed as where h is an increasing and convex function such that h(−∞) = −∞ and h(a) = 0, and where I[•] denotes the indicator function, equal to 1 if the event appearing in the brackets is realized and to 0 otherwise.As an example, distribution functions of the form with α ≤ 1 are log-convex.In particular, for two homogeneous samples with the same marginal distribution functions, that is when F = G, Corollary 3.3 (i) tells us that the variability (in terms of the dispersive order) of the largest statistics is increasing in the dependence parameter θ.
(ii) Likewise, if we consider two Clayton copulas Ĉφ 1 and Ĉφ 2 for the survival copulas with parameters 0 < θ 1 ≤ θ 2 , respectively, then we know from Corollary 3.3 (ii) that if Ḡ is log-convex and X disp Y , then X 1:n disp Y 1:n .Well-known distributions have log-convex survival functions Ḡ, as shown in [19].Let us mention the Pareto distribution, the Gamma distribution with density function f (x) = x c−1 exp(−c) Γ(c) and 0 < c < 1 as well as the Weibull distribution with density function f (x) = cx c−1 exp(−x c ) and 0 < c < 1.
The log-convexity condition on the marginal distribution function G in Proposition 3.2 and Corollary 3.3 is necessary to establish dispersive order inequalities among the largest order statistics, as illustrated in the following example.
which is actually not fulfilled for most of the Archimedean copulas.In particular, if we consider the Clayton copula, we get As a result, k 2 (α) is not monotone, as illustrated in Figure 3.2 for n = 3, θ 1 = 1 and θ 2 = 2. Therefore, U n:n and V n:n cannot be compared with the dispersive order, the reason is that the uniform distribution is log-concave and not log-convex.

It is interesting to notice
, which amounts to requiring ln U n:n disp ln V n:n .Therefore the condition in Proposition 3.2 that δ −1 C X (α)/δ −1 C Y (α) is decreasing in α ∈ [0, 1] can be rewritten as ln U n:n disp ln V n:n .4 Comparison of the largest order statistics with the dispersive order for different sample sizes Let X n = (X 1 , . . ., X n ) be an homogeneous random vector with common distribution function F assumed to be twice differentiable and let X n−1 = (X 1 , . . ., X n−1 ).In this section, we aim to compare the largest order statistics of samples X n and X n−1 with the dispersive order.We denote by δ n and δ n−1 the diagonal sections of the copulas of X n and X n−1 , respectively.
Proof.First, we know that Therefore, we clearly have that X n:n disp X n−1:n−1 if, and only if, n−1 (α)} is a decreasing function.Similar calculations than in the proof of Proposition 3.2 yield where For Archimedean copulas, Proposition 4.1 directly leads to the next result.
The following example illustrates the condition on φ involved in Corollary 4.2.As mentioned in Li and Fang [20] and Mesfioui et al. [21], this condition is satisfied for most of the Archimedean copulas, such as Clayton, Frank and Gumbel copulas.
As already discussed in the previous section for Proposition 3.2 and Corollary 3.3, the log-convexity of F required in Proposition 4.1 and Corollary 4.2 is also crucial here to ensure the ordering of X n:n and X n−1:n−1 with the dispersive order, as revealed by the next example.
Example 4.4.Consider the homogeneous uniform random vector U = (U 1 , . . ., U n ) distributed as the independence copula C φ .The generator of the independence copula, that is φ(t) = e −t , satisfies tφ (t)/φ(t) = −t, so that it is well decreasing in t > 0, as required in Corollary 4.2.However, it is easy to see that the uniform distribution F is not log-convex but log-concave.In this case, the quantile functions of X n−1:n−1 and X n:n are F −1 n−1:n−1 (α) = α   Actually, for a uniform random vector U = (U 1 , . . ., U n ) distributed as C φ , a necessary and sufficient condition on the generator φ to get U n:n disp U n−1:n−1 can be obtained, as shown next.