Stochastic somparisons of order statistics from scaled and ... · Along this line of research, this...

MetrikaDOI 10.1007/s00184-015-0567-3

Stochastic somparisons of order statistics from scaledand interdependent random variables

Chen Li1 · Rui Fang2 · Xiaohu Li3

Received: 4 February 2015© Springer-Verlag Berlin Heidelberg 2015

Abstract This paper studies order statistics from randomvariables following the scalemodel. In the presence of the Archimedean copula or survival copula for the randomvariables, we obtain the usual stochastic order of the sample extremes and the secondsmallest order statistic, the dispersive order and the star order of the sample extremes.

Keywords Archimedean copula · DHR · DRHR · Majorization · Stochastic order

1 Introduction

Let Xk:n be the k-th smallest order statistic of random variables X1, . . . , Xn . In relia-bility theory, Xk:n characterizes the lifetime of a (n − k + 1)-out-of-n system, whichworks if at least n − k + 1 of the n components function normally. Particularly, X1:n ,Xn:n and X2:n represent the lifetimes of the series, parallel and fail-safe systems,respectively. Also, the sample extremes and the second smallest order statistic havenice applications in auction theory. For example, the maximum and minimum definethe final price of the first-price sealed-bid auction and the first-price procurementauction, and the second smallest order statistic is exactly the final price in the second-price procurement auction. One may refer to Milgrom (2004), Menezes and Monteiro(2005) and Krishna (2010) for comprehensive expositions of auction theory, and werefer readers to Li (2005) and Fang and Li (2015) for applications of order statistics in

B Xiaohu [email protected]; [email protected]

1 School of Mathematical Sciences, Xiamen University, Xiamen 361005, Fujian, China

2 Department of Mathematics, Shantou University, Shantou 515063, Guangdong, China

3 Department ofMathematical Sciences, Stevens Institute of Technology, Hoboken, NJ 07030, USA

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C. Li et al.

auction theory. Besides, order statistics play a role in statistical inference, operationresearch, economics and many other applied probability fields. There are consider-able studies on order statistics during the past several decades and a large numberof which are on stochastic comparisons of order statistics from heterogeneous andhomogeneous samples. Due to the complexity of the distribution theory, most existingresearch assumes the mutual independence among concerned random variables. Forcomprehensive references onemay refer to David andNagaraja (2003), Kochar (2012)and Balakrishnan and Zhao (2013).

Let F(x, λ) be a distribution function with parameter λ and F(x, λ) = 1− F(x, λ)

be the corresponding survival function. Denote In = {1, . . . , n}, λ = (λ1, . . . , λn)

and μ = (μ1, . . . , μn). For two groups of mutually independent random variablesXi ∼ F(x, λi ), i ∈ In and Yi ∼ F(x, μi ), i ∈ In , (1971, Theorems 2.8 and 2.9)first showed that, if F(x, λ) (F(x, λ)) is differentiable, monotone and log-convex withrespect to λ ≥ 0 for x ≥ 0, then

λm� μ �⇒ Xk:n ≤st (≥st) Yk:n, k ∈ In, (1.1)

and if F(x, λ) (F(x, λ)) is differentiable and log-concave in λ ≥ 0 for x ≥ 0, then

λm� μ �⇒ Xn:n ≤st Yn:n (X1:n ≥st Y1:n), (1.2)

where ‘m�’ and ‘≤st’ respectively denote themajorization and the usual stochastic order

(see Sect. 2 for their definitions).The randomvariables X1, . . . , Xn are said to follow the scalemodel if Xi ∼ F(λi x)

for i ∈ In , i.e., Xi has the distribution function F(λi x), where F is the baselinedistribution and λi ’s are the scale parameters. Many commonly used distributionshave scale parameters, for example, exponential distribution E(λ) with density λe−λx

for x > 0, λ > 0, Weibull distribution W(α, λ) with density αxα−1λαe−(λx)α forx > 0, α > 0, λ > 0, Lomax distribution L(α, λ) with density αλ(1 + λx)−α−1 forx > 0, α > 0, λ > 0, Fréchet distribution F(α, λ) with density αλ−αx−α−1e−(λx)−α

for x > 0, α > 0, λ > 0, and Gamma distribution G(α, λ)with density λα

�(α)xα−1e−λx

for x > 0, α > 0, λ > 0 all have the scale parameter λ.For the exponential samples Xi ∼ E(λi ), i ∈ In , and Yi ∼ E(μi ), i ∈ In , both

mutually independent, Pledger and Proschan (1971, Corollary 2.7) showed that

λm� μ �⇒ X1:n

st= Y1:n and Xk:n ≤st Yk:n, k = 2, . . . , n, (1.3)

where ‘st=’ means both sides have a common distribution. Also for Xi ∼ E(λi ), i ∈ In ,

and Zi ∼ E(λ), i ∈ In , both mutually independent, Dykstra et al. (1997, Theorem2.1) verified that

λ = 1

n

n∑

i=1

λi �⇒ Xn:n ≥disp Zn:n,

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Stochastic comparisons of order statistics from scaled…

where ‘≤disp’ denotes the dispersive order (see Sect. 2 for the definition). Later on,Khaledi and Kochar (2000, Theorems 2.2 and 2.1) strengthened the result on themaximums of the exponential samples in (1.2) as

λp� μ �⇒ Xn:n ≤st Yn:n, (1.4)

and showed that

λ =( n∏

i=1

λi

) 1n �⇒ Xn:n ≥disp Zn:n,

where ‘p�’ denotes the p-larger order (see Sect. 2 for the definition).

As for Weibull samples Xi ∼ W(α, λi ), i ∈ In and Yi ∼ W(α, μi ), i ∈ In , bothmutually independent, Khaledi and Kochar (2006, Corollaries 3.2 and 3.1) furtherproved that

λp� μ �⇒ Xn:n ≤st Yn:n, (1.5)

and for Xi ∼ W(α, λi ), i ∈ In and Zi ∼ W(α, λ), i ∈ In , bothmutually independent,

λ =( n∏

i=1

λi

) 1n �⇒ Xn:n ≥disp Zn:n, for 0 < α ≤ 1.

In the case of Gamma samples Xi ∼ G(α, λi ), i ∈ In , and Yi ∼ G(α, μi ), i ∈ In ,both mutually independent, Sun and Zhang (2005, Theorem 1.2) proved that

λm� μ �⇒ X1:n ≥st Y1:n and Xn:n ≤st Yn:n, for α ≥ 1, (1.6)

which actually follows immediately from (1.2) by setting the distribution functionF(x, λ) = ∫ x

0λα

�(α)tα−1e−λt dt for α ≥ 1.

As is known, both the exponential and the Weibull distributions follow the scalemodel and are of decreasing proportional reversed hazard rate (DPRHR) (see Sect. 2for the definition). For mutually independent Xi ∼ F(λi x), i ∈ In and Yi ∼ F(μi x),i ∈ In , Khaledi et al. (2011, Theorem 3.2) strengthened (1.4) and (1.5) to

λp� μ �⇒ Xn:n ≤st Yn:n whenever F is DPRHR. (1.7)

Recently, some authors paid their attention to comparing order statistics of depen-dent samples. For instance, Navarro and Spizzichino (2010) studied stochastic ordersof series and parallel systems with components’ lifetimes sharing a common copula,Rezapour and Alamatsaz (2014) obtained stochastic orders on order statistics fromsamples with different survival Archimedean copulas, and Li and Fang (2015) investi-gated stochastic orders of the maximums of samples having proportional hazards andcoupled by Archimedean copula. In particularly, for Weibull samples Xi ∼ W(α, λi ),

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C. Li et al.

i ∈ In , and Yi ∼ W(α, μi ), i ∈ In , both sharing a common Archimedean survivalcopula with generator ψ , Li and Li (2015, Corollary 4.3) showed that

λ �w μ �⇒ X1:n ≥st Y1:n, for log-convex ψ and α ≥ 1, (1.8)

λ �w μ �⇒ X1:n ≤st Y1:n, for log-concave ψ and α ≤ 1, (1.9)

and for Xi ∼ W(α, λi ), i ∈ In and Zi ∼ W(α, λ), i ∈ In , both sharing a commonArchimedean survival copula with generatorψ , Li and Li (2015, Theorem 4.4) provedthat

λα ≤ (≥)1

n

n∑

k=1

λαk �⇒ X1:n ≤disp (≥disp) Z1:n

whenever lnψ and ψ lnψψ ′ are both convex (concave), where ‘�w’ and ‘�w’ denote the

weak submajorization and supermajorization (see Sect. 2 for definitions), respectively.Along this line of research, this note further devotes to studying stochastic orders

on sample extremes and the second smallest order statistic from random variablesfollowing the scale model and coupled by Archimedean copulas or survival copulas.The rest of this paper is organized as follows: Sect. 2 recalls those important conceptsconcerned with our studies and some related results in the literature. In Sect. 3 wepresent several useful lemmas that will be utilized to develop the main theorems inthe sequel. Section 4 investigates the usual stochastic order on sample extremes andthe second smallest order statistic, and Sect. 5 studies the dispersive order and the starorder on sample extremes.

For convenience, from now on we denote the sets R = (−∞,+∞), R+ =(0,+∞), In = {1, . . . , n}, the real vectors λ = (λ1, . . . , λn), μ = (μ1, . . . , μn),1 = (1, . . . , 1), and the random vectors X = (X1, . . . , Xn), Y = (Y1, . . . ,Yn)Z = (Z1, . . . , Zn). Throughout this note, all random variables are assumed to be non-negative and absolutely continuous, and the terms terms increasing and decreasingmean nondecreasing and nonincreasing, respectively.

2 Preliminaries

For ease of reference, in this section we recall some related concepts and presentseveral lemmas that will be used in deriving the main results in the sequel.

A distribution function F with hazard rate s(·) and reversed hazard rate r(·) is saidto be of

(i) Decreasing reversed hazard rate (denoted as DRHR) if r(x) is decreasing;(ii) Increasing/decreasing hazard rate (denoted as IHR/DHR) if s(x) is increas-

ing/decreasing;(iii) Decreasing proportional reversed hazard rate (denoted as DPRHR) if xr(x) is

decreasing;(iv) Increasing proportional hazard rate (denoted as IPHR) if xs(x) is increasing.

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For more on the above aging properties, one may refer to Barlow and Proschan (1975),Marshall and Olkin (2007), Righter et al. (2009) and Li and Fang (2015).

For two random variables X and Y with distribution functions F and G, denoteF−1 and G−1 their respective right continuous inverses, and F and G their respectivesurvival functions. Then, X is said to be smaller than Y in the

(i) Usual stochastic order (denoted as X ≤st Y ) if F(t) ≤ G(t) for all t ;(ii) Dispersive order (denoted as X ≤disp Y ) if F−1(β) − F−1(α) ≤ G−1(β) −

G−1(α) for all 0 < α ≤ β < 1;(iii) Star order (denoted as X ≤∗ Y ) if G−1

(F(t)

)/t increases in t ≥ 0.

Formore on these stochastic orders onemay refer toMüller and Stoyan (2002), Shakedand Shanthikumar (2007) and Li and Li (2013).

For two real vectors x = (x1, . . . , xn) and y = (y1, . . . , yn) in Rn , denote x(1) ≤

· · · ≤ x(n) the increasing arrangement of x1, . . . , xn . Then, x is said to be

(i) Majorized by y (denoted as xm� y) if

∑ni=1 xi = ∑n

i=1 yi and∑ j

i=1 x(i) ≥∑ j

i=1 y(i) for j ∈ In−1;(ii) Weakly submajorized by y (denoted as x �w y) if

∑ni= j x(i) ≤ ∑n

i= j y(i) forj ∈ In ;

(iii) Weakly supermajorized by y (denoted as x �w y) if∑ j

i=1 x(i) ≥ ∑ ji=1 y(i) for

j ∈ In .Also a vector y ∈ R

n+ is said to be p-larger than x ∈ Rn+ (denoted as x

p� y) if∏ j

i=1 x(i) ≥ ∏ ji=1 y(i) for j ∈ In . It is well-known that, for x, y ∈ R

n+,

xp� y ⇐� x �w y ⇐� x

m� y �⇒ x �w y.

A real function h defined on A ⊆ Rn is said to be Schur-convex (Schur-concave)

on A if

xm� y on A �⇒ h(x) ≤ (≥) h( y).

Clearly, h is Schur-concave on A if and only if −h is Schur-convex. For more detailson the above partial orders of real vectors, Schur-convexity and Schur-concavity werefer readers to Bon and Paltanea (1999) and Marshall et al. (2011).

The following three lemmas concerning majorization and Schur-convex or Schur-concave functions are useful in developing our main results in the sequel.

Lemma 2.1 (Marshall et al. (2011), Theorem 3.A.4) For an open interval I ⊆ R, acontinuously differentiable h : I n → R is Schur-convex if and only if h is symmetricon I n and

(xi − x j )(∂ h(x)

∂xi− ∂ h(x)

∂x j

)≥ 0, for all 1 ≤ i �= j ≤ n and x ∈ I n .

Lemma 2.2 (Marshall et al. (2011), Theorem 3.A.8)For a real function h onA ⊆ Rn,

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C. Li et al.

(i) x �w y implies h(x) ≤ h( y) if and only if h is increasing and Schur-convex onA, and

(ii) x �w y implies h(x) ≤ h( y) if and only if h is decreasing and Schur-convex onA.

Lemma 2.3 (Khaledi and Kochar (2002), Lemma 2.1) For a function h : Rn+ �→ R,

xp� y �⇒ h(x) ≤ h( y)

if and only if h(ea1 , . . . , ean ) is decreasing in ai = ln xi , i ∈ In and Schur-convex in(a1, . . . , an).

For a random vector X = (X1, . . . , Xn) with the joint distribution function F ,joint survival function F , univariate marginal distribution functions F1, . . . , Fn andunivariate survival functions F1, . . . , Fn , if there exists someC : [0, 1]n �→ [0, 1] andC : [0, 1]n �→ [0, 1] such that, for all xi , i ∈ In ,

F(x1, . . . , xn) = C(F1(x1), . . . , Fn(xn)

),

F(x1, . . . , xn) = C(F1(x1), . . . , Fn(xn)

),

then C and C are called as the copula and survival copula of X , respectively.A real function ψ is n-monotone on (a, b) ⊆ (−∞,+∞) if (−1)n−2ψ(n−2) is

decreasing and convex in (a, b) and (−1)kψ(k)(x) ≥ 0 for all x ∈ (a, b), k =0, 1, . . . , n − 2. For a n-monotone function ψ : [0,+∞) → [0, 1] with ψ(0) = 1and lim

x→+∞ ψ(x) = 0, then

Cψ(u1, . . . , un) = ψ(ψ−1(u1) + · · · + ψ−1(un)

), for all ui ∈ [0, 1], i ∈ In,

is called an Archimedean copula with generator ψ . For convenience, we denote φ =ψ−1 = sup{x ∈ R : ψ(x) > u}, the right continuous inverse. Archimedean copulascover a wide range of dependence structures including the independence copula withgenerator e−t . For more on Archimedean copulas, readers may refer to Nelsen (2006)and McNeil and Nešlehová (2009).

At the end of this section, we also recall two useful lemmas, which play a role in theproofs of theorems in Sects. 4 and 5. One reviewer points out that the two-dimensionalcase of Lemma 2.4 below had been proved in Theorem 4.4.2 of Nelsen (2006).

Lemma 2.4 (Li and Fang (2015), Lemma A.1) For two n-dimensional Archimedeancopulas Cψ1(u) and Cψ2(u), if φ2 ◦ ψ1 is super-additive, then Cψ1(u) ≤ Cψ2(u) forall u ∈ [0, 1]n.Lemma 2.5 (Cebyšev’s inequality) For two real vectors (a1, . . . , an) and(b1, . . . , bn),

(1

n

n∑

k=1

ak

)(1

n

n∑

k=1

bk

)≤ 1

n

n∑

k=1

akbk

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whenever a1 ≤ · · · ≤ an and b1 ≤ · · · ≤ bn, or a1 ≥ · · · ≥ an and b1 ≥ · · · ≥ bn.

3 Several useful lemmas

For ease of reference, by X ∼ S(F,λ, ψ) and X ∼ S(F,λ, ψ)wedenote randomvari-ables X1, . . . , Xn coupled respectively by the Archimedean copula and Archimedeansurvival copula with generator ψ and following the scale model with baseline distri-bution function F and scale parameter vector λ. Let f (·), s(·) and r(·) be the densityfunction, the hazard rate and the reversed hazard rate of the baseline distribution F ,respectively.

Denote, for x ≥ 0, λ ∈ Rn+,

J1(λ; x, ψ) = ψ

( n∑

k=1

φ(F(λk x)

)),

J2(λ; x, ψ) = 1 − ψ

( n∑

k=1

φ(F(λk x)

)),

J3(λ; x, ψ) =n∑

l=1

ψ

( ∑

k �=l

φ(F(λk x)

)) − (n − 1)ψ

( n∑

k=1

φ(F(λk x)

)).

Evidently, Ji (λ; x, ψ) is symmetric with respect to λ and (ln λ1, . . . , ln λn), i =1, 2, 3.

In this section, we present several lemmas that play an important role in developingthe theorems in the coming Sects. 4 and 5.

Lemma 3.1 If s(x) is increasing and log-convex, then x[ln s(x)]′ is increasing.Proof Since s(x) and hence ln s(x) is increasing, it holds that [ln s(x)]′ ≥ 0. In viewof the log-convexity of s(x), we conclude that [ln s(x)]′ is increasing. So, x[ln s(x)]′is increasing. ��Lemma 3.2 J1(λ; x, ψ) is decreasing in ln λi for i ∈ In, and the log-convexity ofψ along with the IPHR property of F implies that J1(λ; x, ψ) is Schur-concave withrespect to (ln λ1, . . . , ln λn).

Proof For z > 0, let

η1(z, x) = zxs(zx)ψ

(φ(F(zx)

))

ψ ′(φ(F(zx)

)) .

Since ψ is n-monotone, we have ψ ′(x) ≤ 0 for x ≥ 0 and hence the partial derivativeof J1(λ; x, ψ) with respect to ln λi is

∂ J1(λ; x, ψ)

∂ ln λi= −ψ ′

( n∑

k=1

φ(F(λk x)

))η1(λi , x) ≤ 0, for i ∈ In .

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C. Li et al.

That is, J1(λ; x, ψ) is decreasing in ln λi for i ∈ In . Furthermore, for i �= j ,

∂ J1(λ; x, ψ)

∂ ln λi− ∂ J1(λ; x, ψ)

∂ ln λ j

= −ψ ′( n∑

k=1

φ(F(λk x)

))[η1(λi , x) − η1(λ j , x)

].

The log-convexity ofψ implies thatψ/ψ ′ is decreasing. Since φ(F(zx)) is increasing

in z > 0, then ψ(φ(F(zx)))ψ ′(φ(F(zx)))

is decreasing in z > 0. From the increasing property of

xs(x) it follows that zxs(zx) is increasing in z > 0, which in turn implies that η1(z, x)is decreasing in z > 0. Consequently, it holds that, for i �= j ,

(ln λi − ln λ j )

(∂ J1(λ; x, ψ)

∂ ln λi− ∂ J1(λ; x, ψ)

∂ ln λ j

)≤ 0

whenever ψ is log-convex and xs(x) is increasing. Now, the desired result followsimmediately from Lemma 2.1. ��

Lemma 3.3 J1(λ; x, ψ) is decreasing in λi for i ∈ In.(i) The log-convexity of ψ along with the IHR property of F implies that J1(λ; x, ψ)

is Schur-concave with respect to λ;(ii) The log-concavity ofψ along with the DHR property of F implies that J1(λ; x, ψ)

is Schur-convex with respect to λ.

Proof Since the logarithm is strictly increasing, it follows from Lemma 3.2 thatJ1(λ; x, ψ) is decreasing in λi for i ∈ In .

For z > 0, let

η2(z, x) = s(zx)ψ

(φ(F(zx)

))

ψ ′(φ(F(zx)

)) .

Then, for i �= j ,

∂ J1(λ; x, ψ)

∂λi− ∂ J1(λ; x, ψ)

∂λ j

= −xψ ′( n∑

k=1

φ(F(λk x)

))[η2(λi , x) − η2(λ j , x)

].

Since ψ is log-convex (log-concave) and φ(F(zx)) is increasing in z > 0, then

ψ/ψ ′ is decreasing (increasing), and thus ψ(φ(F(zx)))ψ ′(φ(F(zx)))

is decreasing (increasing) in

z > 0. Moreover, the increasing (decreasing) property of s(x) implies that s(zx) is

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increasing (decreasing) in z > 0, and thus η2(z, x) is decreasing (increasing) in z > 0.Consequently, it holds that, for i �= j ,

(λi − λ j )

(∂ J1(λ; x, ψ)

∂λi− ∂ J1(λ; x, ψ)

∂λ j

)≤ 0

whenever ψ is log-convex and s(x) is increasing, and

(λi − λ j )

(∂ J1(λ; x, ψ)

∂λi− ∂ J1(λ; x, ψ)

∂λ j

)≥ 0

wheneverψ is log-concave and s(x) is decreasing. Hence, we obtain the desired resultsdue to Lemma 2.1. ��Lemma 3.4 J2(λ; x, ψ) is decreasing in ln λi for i ∈ In, and the log-convexity ofψ along with DPRHR property of F implies that J2(λ; x, ψ) is Schur-convex withrespect to (ln λ1, . . . , ln λn).

Proof For z > 0, let

η3(z, x) = zxr(zx)ψ

(φ(F(zx)

))

ψ ′(φ(F(zx)

)) .

Since ψ is n-monotone, we have ψ ′(x) ≤ 0 for x ≥ 0 and hence

∂ J2(λ; x, ψ)

∂ ln λi= −ψ ′

( n∑

k=1

φ(F(λk x)

))η3(λi , x) ≤ 0, for i ∈ In .

That is, J2(λ; x, ψ) is decreasing in ln λi for i ∈ In . Furthermore, for i �= j ,

∂ J2(λ; x, ψ)

∂ ln λi− ∂ J2(λ; x, ψ)

∂ ln λ j

= −ψ ′( n∑

k=1

φ(F(λk x)

))[η3(λi , x) − η3(λ j , x)

].

Note that the log-convexity of ψ implies the decreasing property of ψ/ψ ′. Sinceφ(F(zx)) is decreasing in z > 0, then ψ(φ(F(zx)))

ψ ′(φ(F(zx))) is increasing in z > 0. Also the

decreasing property of xr(x) implies that zxr(zx) is decreasing in z > 0, and thusη3(z, x) is increasing in z > 0. Consequently, it holds that, for i �= j ,

(ln λi − ln λ j )

(∂ J2(λ; x, ψ)

∂ ln λi− ∂ J2(λ; x, ψ)

∂ ln λ j

)≥ 0

whenever ψ is log-convex and xr(x) is decreasing. Then the desired result followsimmediately from Lemma 2.1. ��

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C. Li et al.

Lemma 3.5 J2(λ; x, ψ) is decreasing in λi for i ∈ In, and the log-convexity of ψ

along with the DRHR property of F implies that J2(λ; x, ψ) is Schur-convex withrespect to λ.

Proof Since the logarithm is strictly increasing, it follows from Lemma 3.4 thatJ2(λ; x, ψ) is decreasing in λi for i ∈ In .

For z > 0, let

η4(z, x) = r(zx)ψ

(φ(F(zx)

))

ψ ′(φ(F(zx)

)) .

Then, for i �= j ,

∂ J2(λ; x, ψ)

∂λi− ∂ J2(λ; x, ψ)

∂λ j

= −xψ ′( n∑

k=1

φ(F(λk x)

))[η4(λi , x) − η4(λ j , x)

].

Since ψ is log-convex and φ(F(zx)) is decreasing in z > 0, by proof of Lemma 3.4,we have ψ(φ(F(zx)))

ψ ′(φ(F(zx))) is increasing in z > 0. Note that the decreasing property ofr(x) implies that r(zx) is decreasing in z > 0, thus η4(z, x) is increasing in z > 0.Consequently, it holds that, for i �= j ,

(λi − λ j )

(∂ J2(λ; x, ψ)

∂λi− ∂ J2(λ; x, ψ)

∂λ j

)≥ 0

whenever ψ is log-convex and r(x) is decreasing, which completes the proof byapplying Lemma 2.1 directly. ��Lemma 3.6 J3(λ; x, ψ) is decreasing in λi for i ∈ In, and the log-concavity ofψ along with the DHR property of F implies that J3(λ; x, ψ) is Schur-convex withrespect to λ.

Proof For 1 ≤ i �= j ≤ n, let

η5(z, x) = (n − 1)ψ ′( n∑

k=1

φ(F(λk x)

)) −∑

l /∈{i, j}ψ ′

( ∑

k �=l

φ(F(λk x)

))

−ψ ′(

φ(F(zx)

) +∑

k /∈{i, j}φ(F(λk x)

)).

Since ψ is n-monotone, it holds that ψ ′(x) ≤ 0 for x ≥ 0 and ψ ′ is increasing. Sinceφ(F(zx)) is increasing in z > 0, the function η5(z, x) is decreasing in z > 0. In viewof φ(x) ≥ 0 for x ∈ [0, 1], we have

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ψ ′( n∑

k=1

φ(F(λk x)

)) ≥ ψ ′( ∑

k �=l

φ(F(λk x)

)), for l ∈ In,

and then

(n − 1)ψ ′( n∑

k=1

φ(F(λk x)

)) ≥∑

l �=i

ψ ′(∑

k �=l

φ(F(λk x)

)), for i ∈ In .

As a result, it holds that

η5(λi , x) = (n − 1)ψ ′( n∑

k=1

φ(F(λk x)

)) −∑

l /∈{i, j}ψ ′

( ∑

k �=l

φ(F(λk x)

))

−ψ ′(∑

k �= j

φ(F(λk x)

))

= (n − 1)ψ ′( n∑

k=1

φ(F(λk x)

)) −∑

l �=i

ψ ′( ∑

k �=l

φ(F(λk x)

))

≥ 0, for any x ≥ 0 and i ∈ In .

Hence,

∂ J3(λ; x, ψ)

∂λi= xη2(λi , x)η5(λi , x) ≤ 0, for i ∈ In .

That is, J3(λ; x, ψ) is decreasing in λi for i ∈ In . Moreover, for 1 ≤ i �= j ≤ n, itholds that

∂ J3(λ; x, ψ)

∂λi− ∂ J3(λ; x, ψ)

∂λ j

= x[η2(λi , x)η5(λi , x) − η2(λ j , x)η5(λ j , x)

].

Note that ψ is log-concave, s(x) is decreasing and φ(F(zx)) is increasing in z > 0.By the proof of Lemma 3.3, η2(z, x) and hence η2(z, x)η5(z, x) is increasing in z > 0.Consequently, we have, for 1 ≤ i �= j ≤ n,

(λi − λ j )

(∂ J3(λ; x, ψ)

∂λi− ∂ J3(λ; x, ψ)

∂λ j

)≥ 0

whenever ψ is log-concave and s(x) is decreasing, and by Lemma 2.1 this completesthe proof. ��

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C. Li et al.

4 Usual stochastic order

This section studies the usual stochastic order on sample extremes and the secondsmallest order statistic from the scaled samples coupled by Archimedean copulas orsurvival copulas. Recall that X ∼ S(F,λ, ψ) denotes the sample (X1, . . . , Xn) withscale parameter vector λ and the Archimedean survival copula generated by ψ andX ∼ S(F,λ, ψ) denotes (X1, . . . , Xn) with the scale parameter vector λ and theArchimedean copula generated by ψ .

4.1 On sample minimum and the second smallest order statistic

For the scale samples with Archimedean survival copulas, we present here the usualstochastic order on the sample minimum and the second smallest order statistic.

Theorem 4.1 Suppose, for X ∼ S(F,λ, ψ1) and Y ∼ S(F,μ, ψ2), ψ1 or ψ2 is log-convex, and φ1 ◦ ψ2 is super-additive. Then, X1:n ≥st Y1:n if (i) (ln λ1, . . . , ln λn) �w(lnμ1, . . . , lnμn) and F is IPHR, or (ii) λ �w μ and F is IHR.

Proof The sample minimums X1:n and Y1:n have their respective survival functions,for x ≥ 0,

P(X1:n > x) = ψ1

( n∑

k=1

φ1(F(λk x)

)) = J1(λ; x, ψ1), (4.1)

P(Y1:n > x) = ψ2

( n∑

k=1

φ2(F(μk x)

)) = J1(μ; x, ψ2). (4.2)

Let us assume here that ψ1 is log-convex, for the case with log-convex ψ2, the proofcan be completed in the similar manner.

(i) Since xs(x) is increasing, from Lemma 3.2 it follows that −J1(λ; x, ψ1) isSchur-convex with respect to (ln λ1, . . . , ln λn) and increasing in ln λi for i ∈ In .According to Lemma 2.2(i), (ln λ1, . . . , ln λn) �w (lnμ1, . . . , lnμn) implies

− J1(λ; x, ψ1) ≤ −J1(μ; x, ψ1). (4.3)

Since φ1 ◦ ψ2 is super-additive, by Lemma 2.4 we have

J1(μ; x, ψ1) ≥ J1(μ; x, ψ2), (4.4)

and thusJ1(λ; x, ψ1) ≥ J1(μ; x, ψ1) ≥ J1(μ; x, ψ2). (4.5)

(ii) Since s(x) is increasing, according to Lemma 3.3(i), −J1(λ; x, ψ1) is Schur-convex with respect to λ and increasing in λi , i ∈ In . Also, since φ1 ◦ ψ2 is super-additive and λ �w μ, by Lemma 2.2(i) and Lemma 2.4, we reach (4.3), (4.4) againand hence (4.5).

123


As a consequence, due to (4.1) and (4.2) it concludes that P(X1:n > x) ≥ P(Y1:n >

x) for x ≥ 0. That is, X1:n ≥st Y1:n . ��Here, we present an example, illustrating the condition on generators in Theo-

rem 4.1.

Example 4.1 Suppose that X and Y have either of the following two dependencestructures.

(i) Gumbel survival copulas with respective generators

ψ1(x) = e−x1/θ1 , ψ2(x) = e−x1/θ2 , for θ1 ≥ θ2 ≥ 1;

(ii) Archimedean survival copulas with respective generators

ψ1(x) = (x1/θ1 + 1)−1, ψ2(x) = (x1/θ2 + 1)−1, for θ1 ≥ θ2 ≥ 1.

It is not difficult to verify that ψi is log-convex for i = 1, 2. In view of φ1(ψ2(0)) = 0and the convexity ofφ1(ψ2(x)) = xθ1/θ2 , we conclude thatφ1(ψ2(x)) is super-additiveby Proposition 21.A.11 in Marshall and Olkin (2007). ��

According to (1.2), for Xi ∼ F(λi x), i ∈ In , and Yi ∼ F(μi x), i ∈ In , bothmutually independent, it holds that

λm� μ �⇒ X1:n ≥st Y1:n whenever F is I H R.

Also, both the Gamma distribution G(α, λ) and Weibull distribution W(α, λ) followthe scale model and are IHR for α ≥ 1. Note that

(i) for two independent samples, the independence survival copula is theArchimedeansurvival copula with log-convex generator e−x ;

(ii) for two samples share a common Archimedean survival copula with a log-convexgenerator, the conditions on generators in Theorem 4.1 are satisfied;

(iii) the majorization λm� μ implies the submajorization λ �w μ.

Theorem 4.1(ii) partially improves the implication in (1.2) by relaxing the inde-pendence assumption in (1.2) under the scale model, Theorem 4.1(ii) significantlygeneralizes the implication in (1.6) by extending the independent Gamma model of(1.6) to the more general dependent scale model, and Theorem 4.1(ii) also improvesthe result in (1.8) by further generalizing the scenariowhere twoWeibull samples sharea common Archimedean survival copula to the situation in which two scale samplespossess possibly different Archimedean survival copulas.

Theorem 4.2 Suppose, for X ∼ S(F,λ, ψ1) and Y ∼ S(F,μ, ψ2), ψ1 or ψ2 islog-concave and φ2 ◦ ψ1 is super-additive. Then, X1:n ≤st Y1:n if λ �w μ and F isDHR.

123

C. Li et al.

Proof Let us assume here that ψ1 is log-concave, for the case with log-concave ψ2,the proof can be completed in the similar manner.

Note that s(x) is decreasing. From Lemma 3.3(ii) it follows that J1(λ; x, ψ1)

is Schur-convex with respect to λ and decreasing in λi for i ∈ In . According toLemma 2.2(ii), λ �w μ implies

J1(λ; x, ψ1) ≤ J1(μ; x, ψ1).

Since φ2 ◦ ψ1 is super-additive, by Lemma 2.4 we have

J1(μ; x, ψ1) ≤ J1(μ; x, ψ2),

and thus

J1(λ; x, ψ1) ≤ J1(μ; x, ψ1) ≤ J1(μ; x, ψ2).

As a consequence, by (4.1) and (4.2) we conclude that P(X1:n > x) ≤ P(Y1:n > x)for x ≥ 0. That is, X1:n ≤st Y1:n . ��


λm� μ �⇒ X1:n ≤st Y1:n and X2:n ≤st Y2:n whenever F is DHR. (4.6)

Moreover, it is easy to verify thatWeibull distributionW(α, λ) follows the scalemodeland is DHR for 0 < α ≤ 1. Note that (i) φ2 ◦ ψ1 is clearly super-additive and both ψ1and ψ2 are log-concave if two samples have a common Archimedean copula with alog-concave generator ψ , namely ψ1(x) = ψ2(x) = ψ(x), and (ii) the majorization

λm� μ implies the supermajorization λ �w μ. Theorem 4.2 partly generalizes the

implication in (1.1) to two dependent scale samples and further improves that in (1.9)by allowing samples to have possibly different Archimedean survival copulas.

About the conditions of the generators in Theorem 4.2 we present one examplebelow.

Example 4.2 (i) Let X andY have theGumbel-Hougaard survival copulaswith respec-

tive generators ψ1(x) = e1θ1

(1−ex )and ψ2(x) = e

1θ2

(1−ex )for 0 < θ2 ≤ θ1 ≤ 1. It is

easy to verify that ψi is log-concave for i = 1, 2. Since

d2[φ2

(ψ1(x)

)]

dx2= e−x

(θ1

θ2− 1

)[1 +

(θ1

θ2− 1

)e−x

]−2 ≥ 0,

for 0 < θ2 ≤ θ1 ≤ 1, it holds that φ2(ψ1(x)) is convex. In view of φ2(ψ1(0)) = 0, byProposition 21.A.11 of Marshall and Olkin (2007) we have that φ2(ψ1(x)) is super-additive.

123


(ii) Let X and Y have the Archimedean survival copulas with respective generatorsψ1(x) = e1−(1+x)1/θ1 and ψ2(x) = e1−(1+x)1/θ2 for 0 < θ1 ≤ θ2 ≤ 1. It is easy tocheck that ψi is log-concave for i = 1, 2. In observation of

d2[φ2

(ψ1(x)

)]

dx2= θ2

θ1

(θ2

θ1− 1

)(1 + x)

θ2θ1

−2 ≥ 0,

for 0 < θ1 ≤ θ2 ≤ 1, we have the convexity of φ2(ψ1(x)). In view of φ2(ψ1(0)) = 0,we reach the super-additivity of φ2(ψ1(x)) by Proposition 21.A.11 of Marshall andOlkin (2007) again. ��

For scale samples with a common Archimedean survival copula, we also obtain theusual stochastic order on the second smallest order statistic.

Theorem 4.3 For X ∼ S(F,λ, ψ) and Y ∼ S(F,μ, ψ) with log-concave ψ ,X2:n ≤st Y2:n if λ �w μ and F is DHR.

Proof The second smallest order statistics X2:n and Y2:n have their respective survivalfunctions, for x ≥ 0,

P(X2:n > x)

= P(X2:n > x, X1:n ≤ x) + P(X2:n > x, X1:n > x)

=n∑

l=1

P(Xl ≤ x, Xk > x, k �= l) + P(Xk > x, k ∈ In)

=n∑

l=1

[P(Xk > x, k �= l) − P(Xk > x, k ∈ In)

] + P(Xk > x, k ∈ In)

=n∑

l=1

P(Xk > x, k �= l) − (n − 1)P(Xk > x, k ∈ In)

=n∑

l=1

ψ

( ∑

k �=l

φ(F(λk x)

)) − (n − 1)ψ

( n∑

k=1

φ(F(λk x)

))

= J3(λ; x, ψ),

and

P(Y2:n > x) =n∑

l=1

ψ

(∑

k �=l

φ(F(μk x)

)) − (n − 1)ψ

( n∑

k=1

φ(F(μk x)

))

= J3(μ; x, ψ).

Since ψ is log-concave and s(x) is decreasing, due to Lemma 3.6, J3(λ; x, ψ)

is Schur-convex with respect to λ and decreasing in λi for i ∈ In . According toLemma 2.2(ii), λ �w μ implies J3(λ; x, ψ) ≤ J3(μ; x, ψ). As a consequence, weconclude that P(X2:n > x) ≤ P(Y2:n > x), x ≥ 0. That is, X2:n ≤st Y2:n . ��

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C. Li et al.

For n = 3, Ali-Mikhail-Haq survival copula has a log-concave generator ψ(x) =1−θex−θ

, θ ∈ [−2 + √3, 0], and for n = 2, the generator ψ(x) = [0.5(ex + 1)]−1/θ for

θ ∈ (0, 1] is log-concave also. Again, note that λm� μ implies λ �w μ. As for the

second smallest order statistic, due to (4.6) and the log-concave generator e−x of theindependence survival copula, our Theorem 4.3 partially improves the result in (1.1).On the other hand, Theorem 4.3 partially generalizes the implication in (1.3) becausethe exponential distribution has a constant hazard rate.

4.2 On sample maximum

In this subsection we consider a dual model to the one studied in the previous sub-section. We assume that samples have some Archimedean copulas and compare thecorrespondingmaximums.Note that, for X ∼ S(F,λ, ψ1) andY ∼ S(F,μ, ψ2), sam-ple maximums Xn:n and Yn:n have their respective survival functions, for all x ≥ 0,

P(Xn:n > x) = 1 − P(Xk ≤ x, k ∈ In)

= 1 − ψ1

( n∑

k=1

φ1(F(λk x)

))

= J2(λ; x, ψ1),

and

P(Yn:n > x) = 1 − ψ2

( n∑

k=1

φ2(F(μk x)

)) = J2(μ; x, ψ2).

In parallel to Theorems 4.1 and 4.2, we obtain the stochastic comparison on thesample maximums also. Since the following theorem can be verified in a similarmanner to Theorems 4.1 and 4.2 based on Lemmas 2.4, 3.4 and 3.5, we omit the prooffor brevity.

Theorem 4.4 Suppose, for X ∼ S(F,λ, ψ1) and Y ∼ S(F,μ, ψ2), ψ1 or ψ2 is log-

convex, and φ1 ◦ ψ2 is super-additive. Then, Xn:n ≤st Yn:n if (i) λp� μ and F is

DPRHR, or (ii) λ �w μ and F is DRHR.


λm� μ �⇒ Xn:n ≤st Yn:n whenever F is DRHR. (4.7)

In particular, for two independent samples we have ψ1(x) = ψ2(x) = e−x and thus

Theorem 4.4(i) coincides with (1.7). Further, due to λm� μ implying λ �w μ, the

implication in (4.7) follows from Theorem 4.4(ii). So, Theorem 4.4(i) serves as ageneralization of the implication in (1.7) and Theorem 4.4(ii) partially improves thatin (1.2).

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5 Homogeneous and heterogeneous samples

In this section, we switch our focus to the dispersive order and the star order betweenextremes of heterogeneous and homogeneous samples.

5.1 Samples sharing a common Archimedean copula or survival copula

First, we investigate how the baseline distribution and dependence structure impacton the variability of the sample minimum.

Theorem 5.1 Suppose, for X ∼ S(F,λ, ψ) and Z ∼ S(F, λ1, ψ), ψ/ψ ′ is decreas-ing and concave. Then, λ ≤ 1

n

∑nk=1 λk implies X1:n ≤disp Z1:n if (i) x[ln s(x)]′ is

increasing and F is IHR, or (ii) φ(F(x)) is convex, F is both DHR and IPHR, and Fhas a convex proportional hazard rate.

Proof The distribution functions of X1:n and Z1:n are respectively, for x ≥ 0,

F1(x) = 1 − ψ

( n∑

k=1

φ(F(λk x)

)), H1(x) = 1 − ψ

(nφ

(F(λx)

)),

and their respective density functions are

f1(x) = ψ ′( n∑

k=1

φ(F(λk x)

)) n∑

k=1

λk f (λk x)

ψ ′(φ(F(λk x)

)) ,

h1(x) = ψ ′(nφ(F(λx)

)) nλ f (λx)

ψ ′(φ(F(λx)

)) .

Denote L1(x;λ) = F−1(ψ

(1n

n∑k=1

φ(F(λk x)

))). Then, for x ≥ 0,

H−11

(F1(x)

) = 1

λL1(x;λ), (5.1)

and

h1(H−11

(F1(x)

)) = ψ ′( n∑

k=1

φ(F(λk x)

)) nλ f(L1(x;λ)

)

ψ ′(1n

∑nk=1 φ

(F(λk x)

)) .

(i) Since s(x) is increasing, then the increasing property of F−1(ψ(x)) implies thatof F−1(ψ(x))s

(F−1(ψ(x))

). Note that ψ/ψ ′ is decreasing, then, for x ≥ 0,

d

dx

ψ(x)

ψ ′(x)= 1 − ψ(2)(x)ψ(x)

(ψ ′(x))2≤ 0.

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C. Li et al.

From the concavity of ψ/ψ ′ it follows that 1 − ψ(2)(x)ψ(x)(ψ ′(x))2 is decreasing. Hence, for

x ≥ 0,

F−1(ψ(x))s(F−1(ψ(x)

))[1 − ψ(2)(x)ψ(x)

(ψ ′(x))2

]

is decreasing. Since both x[ln s(x)]′ and F−1(ψ(x)) are increasing, then, for x ≥ 0,

F−1(ψ(x))[ f ′(F−1(ψ(x))

)

f(F−1(ψ(x))

) + s(F−1(ψ(x))

)]

is increasing. In view of

d

dx

F−1(ψ(x)) f(F−1(ψ(x))

)

ψ ′(x)

= −1 − F−1(ψ(x))f ′(F−1(ψ(x))

)

f(F−1(ψ(x))

) − F−1(ψ(x))s(F−1(ψ(x))

)ψ(2)(x)ψ(x)

(ψ ′(x))2

= −1 − F−1(ψ(x))

[f ′(F−1(ψ(x))

)

f(F−1(ψ(x))

) + s(F−1(ψ(x))

)]

+ F−1(ψ(x))s(F−1(ψ(x))

)[1 − ψ(2)(x)ψ(x)

(ψ ′(x))2

], for x ≥ 0,

we conclude that ddx

[F−1(ψ(x)) f (F−1(ψ(x)))/ψ ′(x)

]is decreasing. That is,

F−1(ψ(x)) f (F−1(ψ(x)))/ψ ′(x) is concave. Then, for x ≥ 0,

1

n

n∑

k=1

λk x f (λk x)

ψ ′(φ(F(λk x)

)) = 1

n

n∑

k=1

F−1(ψ

(φ(F(λk x)

)))f(F−1

(ψ

(φ(F(λk x)

))))

ψ ′(φ(F(λk x)

))

≤ L1(x;λ) f(L1(x;λ)

)

ψ ′(1n

∑nk=1 φ

(F(λk x)

)) . (5.2)

Since ψ/ψ ′ is decreasing and φ(F(x)) and s(x) both are increasing, then

d

dxφ(F(x)

) = −s(x)ψ

(φ(F(x)

))

ψ ′(φ(F(x)

))

is increasing, i.e., φ(F(x)) is convex. Note thatφ(F(x)) is increasing. λ ≤ 1n

∑nk=1 λk

implies

1

n

n∑

k=1

φ(F(λk x)

) ≥ φ

(F

(x

n

n∑

k=1

λk

))≥ φ

(F(λx)

), for x ≥ 0,

123


and thus the increasing property of F−1(ψ(x)

)leads to

L1(x;λ) = F−1(

ψ

(1

n

n∑

k=1

φ(F(λk x)

)))≥ λx . (5.3)

Consequently, it holds that

h1(H−11

(F1(x)

)) − f1(x)

= nψ ′( n∑

k=1

φ(F(λk x)

))[λ f

(L1(x;λ)

)

ψ ′(1n

∑nk=1 φ

(F(λk x)

)) − 1

n

n∑

k=1

λk f (λk x)

ψ ′(φ(F(λk x)

))]

sgn= 1

n

n∑

k=1

λk x f (λk x)

ψ ′(φ(F(λk x)

)) − λx f(L1(x;λ)

)

ψ ′(1n

∑nk=1 φ

(F(λk x)

))

≤ L1(x;λ) f(L1(x;λ)

)

ψ ′(1n

∑nk=1 φ

(F(λk x)

)) − λx f(L1(x;λ)

)

ψ ′(1n

∑nk=1 φ

(F(λk x)

))

sgn= λx − L1(x;λ)

≤ 0, for x ≥ 0, (5.4)

where ‘sgn= ’ means both sides have the same sign, and respectively the first and the

second inequalities follow from (5.2) and (5.3).(ii) From the concavity of ψ/ψ ′ it follows that, for x ≥ 0,

ψ(1n

∑nk=1 φ

(F(λk x)

))

ψ ′(1n

∑nk=1 φ

(F(λk x)

)) ≥ 1

n

n∑

k=1

F(λk x)

ψ ′(φ(F(λk x)

)) . (5.5)

Note thatφ(F(x)) is increasing and convex.Due to Jensen’s inequalityλ ≤ 1n

∑nk=1 λk

implies

1

n

n∑

k=1

φ(F(λk x)

) ≥ φ

(F

(x

n

n∑

k=1

λk

))≥ φ

(F(λx)

), for x ≥ 0,

and thus the increasing property of F−1(ψ(x)) leads to

L1(x;λ) = F−1(

ψ

(1

n

n∑

k=1

φ(F(λk x)

)))≥ λx . (5.6)

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C. Li et al.

In view of (5.6), λ ≤ 1n

∑nk=1 λk , the decreasing property of s(x), the increasing

property and the convexity of xs(x), we have, for x ≥ 0,

λxs(L1(x;λ)

) ≤ λxs(λx) ≤ 1

n

n∑

k=1

λk x · s(1n

n∑

k=1

λk x)

≤ 1

n

n∑

k=1

λk xs(λk x). (5.7)

Since ψ/ψ ′ is decreasing, and xs(x) and φ(F(x)) are increasing, then according toLemma 2.5, we have, for x ≥ 0,

1

n

n∑

k=1

λk xs(λk x)ψ(φ(F(λk x)

))

ψ ′(φ(F(λk x)

)) − 1

n

n∑

k=1

λk xs(λk x) · 1n

n∑

k=1

ψ(φ(F(λk x)

))

ψ ′(φ(F(λk x)

)) ≤ 0.

(5.8)Consequently, by (5.4) it holds that, for x ≥ 0,

h1(H−11

(F1(x)

)) − f1(x)

sgn= 1

n

n∑

k=1


))

ψ ′(φ(F(λk x)

)) −λxs

(L1(x;λ)

)ψ

(1n

∑nk=1 φ

(F(λk x)

))

ψ ′(1n

∑nk=1 φ

(F(λk x)

))

≤ 1

n

n∑

k=1


))

ψ ′(φ(F(λk x)

)) − λxs(L1(x;λ)

) · 1n

n∑

k=1

F(λk x)

ψ ′(φ(F(λk x)

))

≤ 1

n

n∑

k=1


))

ψ ′(φ(F(λk x)

)) − 1

n

n∑

k=1

λk xs(λk x) · 1n

n∑

k=1

F(λk x)

ψ ′(φ(F(λk x)

))

≤ 0,

where the three inequalities follow from (5.5), (5.7) and (5.8), respectively.Now, we can conclude that f1

(F−11 (x)

) ≥ h1(H−11 (x)

)for all x ∈ (0, 1). By

(3.B.11) in Shaked and Shanthikumar (2007), this invokes X1:n ≤disp Z1:n . ��In the following, we provide an example to illustrate those conditions concerned

with Theorem 5.1.

Example 5.1 (i) Set n = 2, α ≥ 1, 0 < λ ≤ λ1+λ22 and λi > 0 for i = 1, 2. Let

Xi ∼ W(α, λi ), i = 1, 2 and Zi ∼ W(α, λ), i = 1, 2, both have the Archimedeansurvival copula with generator

ψ(x) ={e−2 arctan x , x ≤ 1;x−1e−π/2, x ≥ 1.

It can be verified that

ψ(x)

ψ ′(x)=

{− 1+x22 , x ≤ 1;

−x, x ≥ 1,

123


is decreasing and concave. On the other hand, we have both s(x) = αxα−1 andx[ln s(x)]′ = α − 1 are increasing in x ≥ 0 for α ≥ 1. So, the conditions in Theo-rem 5.1(i) are satisfied.

(ii) Set αθ ≥ 1, 0 < λ ≤ 1n

∑nk=1 λk and λi > 0 for i ∈ In . Let Xi ∼ L(α, λi ),

i ∈ In and Zi ∼ L(α, λ), i ∈ In , both have the Clayton survival copula with generatorψ(x) = (θx + 1)−1/θ . Clearly, ψ(x)/ψ ′(x) = −θx − 1 is decreasing and concave.On the other hand, it is easy to verify that s(x) = α(1 + x)−1 is decreasing andxs(x) = αx(1 + x)−1 is increasing and convex. Since, for αθ ≥ 1,

d2φ(F(x)

)

dx2= α(αθ − 1)(1 + x)αθ−2 ≥ 0,

we have the convexity of φ(F(x)). Therefore, the conditions of Theorem 5.1(ii) aresatisfied also. ��

The next theorem presents a sufficient condition for the star order between the sam-ple minimum of one heterogeneous scale sample and that of the other homogeneousscale sample.

Theorem 5.2 For X ∼ S(F,λ, ψ) and Z ∼ S(F, λ1, ψ) with decreasing and con-cave ψ/ψ ′, X1:n ≤∗ Z1:n if x[ln s(x)]′ is increasing and F is IPHR.

Proof Since xs(x) and x[ln s(x)]′ are increasing, according to the proof of Theo-rem 5.1(i), the decreasing property and the concavity of ψ/ψ ′ imply the concavity

of F−1(ψ(x)) f (F−1(ψ(x)))ψ ′(x) , and thus we reach (5.2) again. Then, by (5.1) we have, for

x ≥ 0,

d

dx

H−11 (F1(x))

x= ψ ′( 1

n

∑nk=1 φ

(F(λk x)

))

λx2 f(L1(x;λ)

)

·[1

n

n∑

k=1

λk x f (λk x)

ψ ′(φ(F(λk x)

)) − L1(x;λ) f(L1(x;λ)

)

ψ ′( 1n

∑nk=1 φ

(F(λk x)

))]

≥ 0.

That is, H−11 (F1(x))/x is increasing in x and hence X1:n ≤∗ Z1:n . ��

The following two corollaries follow immediately from Lemma 3.1 and Theo-rems 5.1(i), 5.2 and thus are presented here with the proofs being omitted for brevity.

Corollary 5.1 For X ∼ S(F,λ, ψ) and Z ∼ S(F, λ1, ψ) with decreasing and con-caveψ/ψ ′,λ ≤ 1

n

∑nk=1 λk implies X1:n ≤disp Z1:n if F is IHRand F has a log-convex

hazard rate.

Corollary 5.2 For X ∼ S(F,λ, ψ) and Z ∼ S(F, λ1, ψ) with decreasing and con-cave ψ/ψ ′, X1:n ≤∗ Z1:n if F is IHR and F has a log-convex hazard rate.

In parallel to randomvariables havingArchimedean survival copulas, we can obtainthe following dual theorem on the maximums of two scale samples with a commonArchimedean copula, and the proof is similar to that of Theorem 5.2 and hence omittedalso.

123

C. Li et al.

Theorem 5.3 For X ∼ S(F,λ, ψ) and Z ∼ S(F, λ1, ψ) with decreasing and con-cave ψ/ψ ′, Xn:n ≤∗ Zn:n if x[ln r(x)]′ is increasing and F is DPRHR.

Here we provide an example of the conditions in Theorem 5.3.

Example 5.2 Let Xi ∼ F(α, λi ), i = 1, 2 and Zi ∼ F(α, λ), i = 1, 2, for α, λ, λi ≥0, i = 1, 2 have the Archimedean copula with generator

ψ(x) ={e−2 arctan x , x ≤ 1;e−1−π/2+1/x , x ≥ 1.

It is plain that

ψ(x)

ψ ′(x)=

{− 1+x22 , x ≤ 1;

−x2, x ≥ 1,

which is decreasing and concave. Also, clearly, xr(x) = αx−α is decreasing in x ≥ 0and x[ln r(x)]′ = −α − 1 is increasing in x ≥ 0. So, the conditions of Theorem 5.3are confirmed. ��

5.2 Samples with different dependence structures

In this subsection, we discuss the dispersive ordering of the sample minimums fromtwo homogenous scale samples with different Archimedean survival copulas. ForZ = (Z1, . . . , Zn), denote Z1:n the minimum of Z.

Theorem 5.4 Suppose, for Z ∼ S(F, λ1, ψ1) and Z ∼ S(F, λ1, ψ2), ψ1 or ψ2 islog-concave and F is DHR. Then, (i) Z1:n ≤disp Z1:n if φ2(ψ1(x)) is convex, and (ii)Z1:n ≥disp Z1:n if φ1(ψ2(x)) is convex.

Proof We only prove the part (i) and the part (ii) can be obtained in a similar manner.The sample minimums Z1:n and Z1:n have their respective distribution functions,

for x ≥ 0,

H1(x) = 1 − ψ1(nφ1

(F(λx)

)), H1(x) = 1 − ψ2

(nφ2

(F(λx)

)),

and the corresponding density functions are

h1(x) = ψ ′1

(nφ1

(F(λx)

)) nλ f (λx)

ψ ′1

(φ1

(F(λx)

)) ,

h1(x) = ψ ′2

(nφ2

(F(λx)

)) nλ f (λx)

ψ ′2

(φ2

(F(λx)

)) .

Also denote, for x ≥ 0,

L2(x; λ) = F−1(ψ1(n−1φ1

(ψ2(nφ2(F(λx)))

))),

L3(x; λ) = F−1(ψ2(n−1φ2

(ψ1(nφ1(F(λx)))

))).

123


Then, for x ≥ 0,

H−11

(H1(x)

) = 1

λL2(x; λ), H−1

1

(H1(x)

) = 1

λL3(x; λ), (5.9)

h1(H−11

(H1(x)

)) = ψ ′1

(nφ1

(F

(L2(x; λ)

))) nλ f(L2(x; λ)

)

ψ ′1

(φ1

(F(L2(x; λ))

)) ,

and

h1(H−11

(H1(x)

)) = ψ ′2

(nφ2

(F

(L3(x; λ)

))) nλ f(L3(x; λ)

)

ψ ′2

(φ2

(F(L3(x; λ))

)) .

First, let us assume that ψ1 is log-concave. Since φ2(ψ1(x)) is increasing and con-

vex, then φ1(ψ2(x)) is increasing and concave and henceψ ′2(x)

ψ ′1(φ1(ψ2(x)))

is decreasing.

In view of φ1(ψ2(0)) = 0, by Proposition 21.A.11 in Marshall and Olkin (2007), wehave φ1(ψ2(x)) is sub-additive, and thus

φ1(F(L2(x; λ))

) = 1

nφ1

(ψ2

(nφ2

(F(λx)

))) ≤ φ1(F(λx)

), x ≥ 0. (5.10)

Then the increasing property of φ1(F(x)) implies λx ≥ L2(x; λ). Since s(x) isdecreasing, then

s(λx) ≤ s(L2(x; λ)

), x ≥ 0. (5.11)

Since ψ1 is log-concave, then ψ ′1/ψ1 is decreasing, and thus (5.10) implies

ψ ′1

(φ1

(F(L2(x; λ))

))

F(L2(x; λ)

) ≥ ψ ′1

(φ1

(F(λx)

))

F(λx), x ≥ 0. (5.12)

In view of nφ2(F(λx)) ≥ φ2(F(λx)) and the decreasing property ofψ ′2(x)

ψ ′1(φ1(ψ2(x)))

, we

haveψ ′2

(nφ2

(F(λx)

))

ψ ′1

(nφ1

(F

(L2(x; λ)

))) ≤ ψ ′2

(φ2

(F(λx)

))

ψ ′1

(φ1

(F(λx)

)) , x ≥ 0. (5.13)

As a consequence, for x ≥ 0,

h1(H−11

(H1(x)

)) − h1(x) = nλ

[ψ ′1

(nφ1

(F

(L2(x; λ)

))) f(L2(x; λ)

)

ψ ′1

(φ1

(F

(L2(x; λ)

)))

−ψ ′2

(nφ2

(F(λx)

)) f (λx)

ψ ′2

(φ2

(F(λx)

))]

123

C. Li et al.

≥ nλF(λx)

[ψ ′1

(nφ1

(F

(L2(x; λ)

))) s(L2(x; λ)

)

ψ ′1

(φ1

(F(λx)

))

−ψ ′2

(nφ2

(F(λx)

)) s(λx)

ψ ′2

(φ2

(F(λx)

))]

≥ 0,

where the first inequality is due to (5.12), and the second one follows from (5.11) and(5.13).

Secondly, we assume that ψ2 is log-concave. Since φ2(ψ1(x)) is convex, thenψ ′1(x)

ψ ′2(φ2(ψ1(x)))

is increasing. In view of φ2(ψ1(0)) = 0, by Proposition 21.A.11 in

Marshall and Olkin (2007) we conclude that φ2(ψ1(x)) is super-additive, and thus

φ2(F(L3(x; λ))

) = 1

nφ2

(ψ1

(nφ1

(F(λx)

))) ≥ φ2(F(λx)

), x ≥ 0. (5.14)

So, the increasing property of φ2(F(x)

)implies λx ≤ L3(x; λ). Due to the decreasing

property of s(x), we have

s(λx) ≥ s(L3(x; λ)

), x ≥ 0. (5.15)

Since ψ2 is log-concave, then ψ ′2/ψ2 is decreasing, and thus (5.14) implies

ψ ′2

(φ2

(F(L3(x; λ))

))

F(L3(x; λ)

) ≤ ψ ′2

(φ2

(F(λx)

))

F(λx), x ≥ 0. (5.16)

In view of nφ1(F(λx)) ≥ φ1(F(λx)) and the increasing property ofψ ′1(x)

ψ ′2(φ2(ψ1(x)))

, we

haveψ ′1

(nφ1

(F(λx)

))

ψ ′2

(nφ2

(F

(L3(x; λ)

))) ≥ ψ ′1

(φ1

(F(λx)

))

ψ ′2

(φ2

(F(λx)

)) , x ≥ 0. (5.17)

As a consequence, it holds that, for x ≥ 0,

h1(H−11

(H1(x)

)) − h1(x) = nλ

[ψ ′2

(nφ2

(F

(L3(x; λ)

))) f(L3(x; λ)

)

ψ ′2

(φ2

(F

(L3(x; λ)

)))

−ψ ′1

(nφ1

(F(λx)

)) f (λx)

ψ ′1

(φ1

(F(λx)

))]

≤ nλF(λx)

[ψ ′2

(nφ2

(F

(L3(x; λ)

))) s(L3(x; λ)

)

ψ ′2

(φ2

(F(λx)

))

−ψ ′1

(nφ1

(F(λx)

)) s(λx)

ψ ′1

(φ1

(F(λx)

))]

≤ 0,

123


where the first inequality follows from (5.16), and the second one follows from (5.15)and (5.17).

Now, we conclude that h1(H−11 (x)

) ≤ h1(H−11 (x)

)for all x ∈ (0, 1), and thus

Z1:n ≤disp Z1:n follows from (3.B.11) of Shaked and Shanthikumar (2007) directly. ��Acknowledgments Authors would like to thank the anonymous reviewer for his/her useful comments,which helped improve the presentation of the earlier version of this manuscript. Dr. Rui Fang’s research issupported by STU Scientific Research Foundation for Talents (NTF15002).

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