L-statistics from multivariate unified

skew-elliptical distributions

R.B. Arellano-Valle1, Ahad Jamalizadeh2, H. Mahmoodian3

and N. Balakrishnan4,5

1 Departamento de Estadıstica, Ponticia Universidad Catolica de Chile,

Santiago, Chile2 Department of Statistics, Shahid Bahonar University, Kerman, Iran

3 Department of Statistics, Yazd University, Yazd, Iran4 Department of Mathematics and Statistics, McMaster University,

Hamilton, Canada5 Department of Statistics, King Abdulaziz University, Jeddah, Saudi Arabia

April 10, 2013

Abstract

We study here the distributions of order statistics and linear combinations of order

statistics from a multivariate unified skew-elliptical distribution. We show that these dis-

tributions can be expressed as mixtures of unified skew-elliptical distributions, and then

use these mixture forms to study some distributional properties and moments.

Keywords: Elliptical distribution; multivariate unified skew-elliptical distributions; Or-

der statistics; Mixture distribution; Linear combination; multivariate unified skew-normal

distribution; multivariate unified skew-t distribution.

1 Introduction

Recently, Arellano-Valle and Genton (2008) presented the density of the maximum of a vector of

dependent random variables. They also derived an expression for the density of the maximum

in the case when the dependent variables jointly have an elliptical density function. Arellano-

Valle and Genton (2007) further presented expressions for densities of order statistics from

an exchangeable multivariate elliptically contoured distribution. This problem was revisited

by Jamalizadeh and Balakrishnan (2010) who proved in general that the distribution of an

1

order statistic from the multivariate elliptical distribution is a mixture of univariate unified

skew-elliptical distributions, thus generalizing the work of Arellano-Valle and Genton (2008).

By using this mixture representation, they also derived the moment generating function and

moments of order statistics, when they exist, through the moment generating function and

moments of the univariate unified skew-elliptical distribution. In addition, they developed

similar results for the joint distribution of linear combinations of order statistics arising from

a multivariate elliptical distribution and expressed them as mixtures of multivariate unified

skew-elliptical distributions (SUE). But, the analogous results for multivariate skew-elliptical

distributions have not been discussed in the literature. In this respect, the results established

here concerning order statistics and linear combinations of order statistics from multivariate

unified skew-elliptical random variables are new and are of very general form.

A first motivation to study the exact distributions of order statistics and linear combinations

of order statistics from dependent random variables comes from a genetic selection problem

in agricultural research, originally considered by Rawlings (1976) and Hill (1976, 1977), and

subsequently by Tong (1990, p. 129). To describe the problem briefly, suppose an agricultural

genetic selection project involves n animals, say pigs, and the top k performers, k < n, are to be

selected for breeding. Let X1, · · · , Xn be the measurements of a certain biological or physical

characteristic on the n animals, such as the body weight or back fat on the animals. The animals

with score X(n−k+1), · · · , X(n), where X(1), · · · , X(n) are the order statistics from X1, . . . , Xn,

are to be selected. However, the assumption of independence of X1, · · · , Xn is not realistic in

this case since the animals under selection are usually genetically related. This is the case, for

example, when the pigs are from the same family and have the same parents. When, in such a

selection problem, the dependent variables follow a multivariate elliptical distribution, then the

exact distributions of order statistics and linear combinations of order statistics can be obtained

from the results of Arellano-Valle and Genton (2007, 2008) and Jamalizadeh and Balakrishnan

(2010). However, the results in this paper enable us to derive the exact distributions of order

statistics and linear combinations of order statistics from a multivariate unified skew-elliptical

distribution. A second motivation comes from vision research where a single measure of visual

acuity is made in each eye, say X1 and X2. A person’s vision total impairment is defined as the

L-statistic TI = 34X(1) + 1

4X(2), where the extremes of visual acuity are X(1) = minX1, X2

and X(2) = maxX1, X2; see Viana (1998) and the references therein for further details. A

bivariate normal distribution is commonly assumed for (X1, X2)T ; however, the assumption of

joint normality for the vector (X1, X2)T is not satisfied in many cases. The results in this paper

would then enable us to derive the exact distribution of TI when (X1, X2)T follows a bivariate

skew-elliptical distribution.

Distributions of order statistics and linear combinations of order statistics from multivariate

normal and multivariate elliptical distributions have been discussed in the literature by many

authors including Gupta and Pillai (1965), Basu and Ghosh (1978), Nagaraja (1982), Balakr-

ishnan (1993), and Wiens et al. (2005). Genc (2006) derived the exact distribution of a linear

combination of order statistics from a bivariate normal distribution, while Arellano-Valle and

Genton (2007, 2008) presented the exact distributions of linear combination of order statistics

2

and of the largest order statistic from multivariate elliptical distributions. Jamalizadeh and

Balakrishnan (2008) derived the exact distributions of order statistics from bivariate skew-

normal and skew-tν distributions. Jamalizadeh et al. (2009) established a recursive scheme for

the evaluation of the cumulative distribution function (cdf) of a linear combination of order

statistics from a bivariate t distribution, while Loperfido (2008) derived the exact distribution

of a linear combination of order statistics from an exchangeable bivariate elliptical distribution.

Jamalizadeh and Balakrishnan (2009) showed that the distributions of order statistics from a

trivariate normal distribution are mixtures of generalized skew-normal distributions, and then

used this mixture form to derive moment generating functions of order statistics and explicit

expressions for the first two moments; they also developed similar results for the case of t

distribution.

The rest of this paper is organized as follows. In Section 2, we first provide a brief review of

the multivariate and univariate classes of SUE distributions, and list some of its main properties

for some special univariate cases. In Section 3, we consider the distribution of order statistics

arising from a multivariate SUE distribution, and show that it is a mixture of univariate SUE

distributions. We then present explicit results for the cases when the kernel distributions are

normal and t. Finally, in Section 4, we discuss the distribution of linear combinations of

order statistics obtained from a multivariate SUE and show that it is indeed a mixture of the

multivariate SUE distributions. Results are then deduced for the special cases of multivariate

SUN and SUT distributions.

The following notation will be used throughout this paper: φ(x) = (2π)−1/2 exp(−1

2x2), x ∈

R, for the standard normal probability density function (pdf), φn(x;µ,Σ) = |Σ|−1/2(2π)−n/2

× exp(−1

2w(x)

), x ∈ Rn, where w(x) = (x−µ)TΣ−1(x−µ), for the pdf of Nn(µ,Σ) (n-variate

normal distribution with mean vector µ and covariance matrix Σ), Φn(· ;µ,Σ) for the cdf of

Nn(µ,Σ) (in both singular and non-singular cases), simply Φn(· ; Σ) for the case when µ = 0,

tn(x;µ,Σ, ν) = |Σ|−1/2Cn,ν(1 + 1

νw(x)

), x ∈ Rn, where Cn,ν = Γ

(ν+n

2

)/Γ(ν2

)(νπ)n/2, for

the pdf of tn(µ,Σ, ν) (n-variate t distribution with location parameter µ, dispersion matrix

Σ, and degrees of freedom ν), Tn(· ;µ,Σ, ν) for the cdf of tn(µ,Σ, ν), simply Tn(· ; Σ, ν)

for the case when µ = 0, fECn(x;µ,Σ, h(n)

)= |Σ|−1/2h(n)(w(x)), x ∈ Rn, for the pdf of

ECn(µ,Σ, h(n)

)(n-variate elliptical distribution with location parameter µ, dispersion matrix

Σ, and density generator function h(n); see e.g. Fang et al., 1990), FECn(· ;µ,Σ, h(n)

)for the

cdf of ECn(µ,Σ, h(n)

), and simply FECn

(· ; Σ, h(n)

)for the case when µ = 0. To include

both singular and non-singular cases, Cambanis et al. (1981) defined a n-variate elliptical

distribution in terms of its characteristic function as exp(itTµ

)ϕ(tTΣt

), t ∈ Rn, for some

characteristic generator function ϕ. In this case, the distribution is denoted by ECn (µ,Σ, ϕ)

and the corresponding cdf by FECn (· ; Σ, ϕ). Furthermore, for r ∈ N, let 1r and Ir denote the

vector of ones and the identity matrix of dimension r, respectively, and let X−i be the vector

obtained from X by deleting its ith component.

3

2 Multivariate and univariate SUE distributions

To better motivate the results developed in this paper, we first provide a brief introduction of

the SUE distributions. We start with the definition of the multivariate SUE distribution that

will used through out this work. Let U and V be two random vectors of dimensions m and n,

respectively, and (U

V

)∼ ECm+n

((η

ξ

),

(Γ ΛT

Ω

), h(m+n)

). (1)

As consequence of (1), we have U|(V = x) ∼ ECm(η + ΛTΩ−1(x−ξ); Γ − ΛTΩ−1Λ, h(m)w(x))

and V ∼ ECn(η,Ω, h(n)), where w(x) = (x−η)Ω−1(x−η) and h(m)w (s) = h(n+m)(s+w)/h(n)(w)

is a conditional density generator function; see, for example, Fang et al. (1990). Then, the

n-dimensional random vector X is said to have the multivariate unified skew-elliptical (SUE)

distribution with parameter θ = (ξ,η,Ω,Γ,Λ) , where ξ ∈ Rn and η ∈ Rm are location vectors,

Ω ∈ Rn×n and Γ ∈ Rm×m are dispersion matrices, Λ ∈ Rn×m is a skewness/shape matrix, and

density generator function h(m+n), denoted by X ∼ SUEn,m(ξ,η,Ω,Γ,Λ, h(m+n)

)or simply

by X ∼ SUEn,m(θ, h(m+n)

), if

Xd= V | (U > 0) . (2)

The density function of X is [see Arellano-Valle and Azzalini (2006), Arellano-Valle et al. (2006)

and Arellano-Valle and Genton ( 2010a)]

gSUEn,m(x;θ, h(m+n)

)=fECn

(x; ξ,Ω, h(n)

)FECm

(η + ΛTΩ−1 (x− ξ) ; Γ−ΛTΩ−1Λ, h

(m)w(x)

)FECm (η; Γ, h(m))

(3)

for x ∈Rn, where w (x) = (x−ξ)T Ω−1 (x−ξ). It reduces to the regular elliptical density

function fECn(x; ξ,Ω, h(n)

)when Λ = 0 and η = 0. It is important to mention here that

we will replace h(m+n) by ϕ(m+n), the characteristic generator function, when we do not know

whether the matrix

(Γ ΛT

Ω

)in (1) is singular or not.

In the normal case, i.e., when h(m+n) (u) = (2π)−(m+n)/2 exp (−u/2) (u ≥ 0), we obtain

the multivariate unified skew-normal distribution (SUN), denoted by X ∼ SUNn,m (θ), with

density function [see Arellano-Valle and Azzalini (2006)]

gSUNn,m (x;θ) =φn (x; ξ,Ω) Φm

(η + ΛTΩ−1 (x−ξ) ; Γ−ΛTΩ−1Λ

)Φm (η; Γ)

.

Furthermore, when X ∼ SUNn,m (θ), the moment generating function (MGF) of X is available

in an explicit form and is given by

MSUNn,m (s;θ) =exp

(ξT s+1

2sTΩs

)Φm

(η + ΛT s; Γ

)Φm (η; Γ)

. (4)

4

In the t case, i.e., when h(m+n) (u) =Γ( ν+m+n

2 )Γ( ν2 )(νπ)

m+n2

(1 + u

ν

)−(ν+m+n)/2(u ≥ 0, ν > 0), we obtain

the multivariate unified skew-t (SUT) distribution, denoted by X ∼ SUTn,m (θ, ν), with density

function [see also Arellano-Valle and Genton (2010a,b)]

gSUTn,m (x;θ, ν)

=

tn(x; ξ,Ω, ν)Tm

(η + ΛTΩ−1 (x− ξ) ;

ν+(x−ξ)T

Ω−1(x−ξ)ν+n

(Γ−ΛTΩ−1Λ

), ν + n

)Tm (η; Γ, ν)

.

Arellano-Valle and Azzalini (2006) presented the marginal and conditional distributions of the

SUN distribution, but not the marginal and conditional distributions of the SUE distribution.

These distributions were developed recently by Arellano-Valle and Genton (2010a) by using,

however, a different but equivalent parametrization of that considered in (2). In the following

lemma, we present these distributions in terms of the parametrization in (2), and they will be

useful for the subsequent sections.

For this purpose, let X1 and X2 be two random vectors of dimensions n1 and n − n1,

respectively, such that (X1

X2

)∼ SUEn,m

(ξ,η,Ω,Γ,Λ, h(m+n)

).

Corresponding to X1 and X2, we consider the following partitions:

ξ =

(ξ1

ξ2

), Ω =

(Ω11 Ω12

Ω22

)and Λ =

(Λ1

Λ2

).

Lemma 1 We then have:

(i) X1 ∼ SUEn1,m

(ξ1,η,Ω11,Γ,Λ1, h

(m+n1));

(ii) For x1 ∈ Rn1 ,

X2 | (X1 = x1) ∼ SUEn−n1,m

(ξ2.1(x1),η2.1(x1),Ω22.1,Γ2.1,Λ2.1, h

(m+n−n1)q1(x1)

), (5)

where

ξ2.1(x1) = ξ2 + Ω21Ω−111 (x1 − ξ1) , η2.1(x1) = η + ΛT

1 Ω−111 (x1 − ξ1) ,

Ω22.1 = Ω22 −Ω21Ω−111 Ω12, Γ2.1 = Γ−ΛT

1 Ω−111 Λ1, Λ2.1 = Λ2 −Ω21Ω

−111 Λ1, (6)

and q1(x1) = (x1 − ξ1)T Ω−111 (x1 − ξ1).

For the t case, we have the following corollary.

Corollary 1 If (X1

X2

)∼ SUTn,m (ξ,η,Ω,Γ,Λ, ν) ,

we then have:

5

(i) X1 ∼ SUTn1,m (ξ1,η,Ω11,Γ,Λ1, ν);

(ii) For x1 ∈ Rn1 ,

X2 | (X1 = x1) ∼ SUTn−n1,m

(ξ2.1(x1),η2.1(x1),Ω22.1(x1),Γ2.1(x1),Λ2.1(x1), ν + n1

),

(7)

where ξ2.1(x1) and η2.1(x1) are as in Lemma 1, and

Ω22.1(x1) =ν + q1(x1)

ν + n1

Ω22.1, Γ2.1(x1) =ν + q1(x1)

ν + n1

Γ2.1, Λ2.1(x1) =ν + q1(x1)

ν + n1

Λ2.1.

(8)

2.1 Univariate SUE distributions

Here, we consider the univariate SUE special class of distributions corresponding to the multi-

variate SUE density in (3) when n = 1 whatever the value of m = 1, 2, . . .. More specifically,

this class correspnds to (1) wherein we have(U

V

)∼ ECm+1

((η

ξ

),

(Γ λ

ω

), h(m+1)

),

where ξ ∈R,η ∈Rm×1, Γ ∈ Rm×m is a positive definite dispersion matrix, ω > 0,λ ∈Rm×1, and

h(m+1) is a density generator function. A real random variable X is said to have a univariate

SUE distribution, denoted by X ∼ SUE1,m

(θ, h(m+1)

), where θ = (ξ,η, ω,Γ,λ), if

Xd= V | (U > 0) . (9)

We further denote by gSUE1,m

(x;θ, h(m+1)

)and GSUE1,m

(x;θ, h(m+1)

)the pdf and cdf of X in

(9), respectively. From the general form in (3), we then have

gSUE1,m

(x;θ, h(m+1)

)=

fEC1

(x; ξ, ω, h(1)

)FECm

(η + λ (x−ξ)

ω; Γ− 1

ωλλT , h

(m)(x−ξ)2ω

)FECm (η; Γ, h(m))

, x∈R,

(10)

where

h(m)a (u) =

h(m+1)(u+ a)

h(1) (a), a, u ≥ 0, (11)

as defined earlier.

2.2 Univariate SUN distributions

In the special case of the normal density generator function h(m+1)(u) = (2π)−(m+1)/2 exp (−u/2)

in (10), we obtain a univariate SUN distribution, denoted by X ∼ SUN1,m (θ), with pdf

gSUN1,m (x;θ) =φ (x; ξ, ω) Φm

(η + λ (x−ξ)

ω; Γ− 1

ωλλT

)Φm (η; Γ)

, x∈R. (12)

6

If MSUN1,m (s;θ) denotes the mgf of X ∼ SUN1,m (θ), then from (4), we obtain for s ∈ R,

MSUN1.m (s;θ) =exp

(ξs+ 1

2ωs2)

Φm (η + λs; Γ)

Φm (η; Γ). (13)

For the derivation of the moments of X in this case, we need the derivatives of MSUN1,m (s;θ)

in (13), for which the following lemma will be useful.

Lemma 2 Let the vectors λ = (λ1, · · · , λm)T and η = (η1, · · · , ηm)T and the m ×m positive

definite matrix Γ = (γij) be partitioned as(λiλ−i

),

(ηiη−i

),

(γii γT−ii

Γ−i−i

),

respectively. Then, we have

∂

∂sΦm (η + λs; Γ) =

m∑i=1

(λi√γii

)φ

(λis+ ηi√

γii

)×Φm−1

((λ−i −

λiγiiγ−ii

)s+

(η−i −

ηiγiiγ−ii

); Γ−i|i

), (14)

where

Γ−i|i = Γ−i−i −γ−iiγ

T−ii

γii. (15)

Theorem 1 If X ∼ SUN1,m (θ), then its mean is given by

E(X) = ξ +1

Φm (η; Γ)

m∑i=1

λi√γii

φ

(ηi√γii

)Φm−1

(η−i −

ηiγiiγ−ii; Γ−i|i

), (16)

where Γ−i|i is as in (15).

2.3 Univariate SUT distributions

In the special case of the t density generator function

h(m+1) (u) =Γ(ν+m+1

2

)Γ(ν2

)(νπ)

m+12

(1 +

u

ν

)−(ν+m+1)/2

, u ≥ 0,

for some ν > 0, the marginal and the conditional density generator functions are

h(m) (u) =Γ(ν+m

2

)Γ(ν2

)(νπ)

m2

(1 +

u

ν

)−(ν+m)/2

, u ≥ 0,

and

h(m)a (u) =

Γ(ν+m+1

2

) (ν+aν+1

)−m/2Γ(ν+1

2

)π (ν + 1)m/2

1 +

(ν + 1

ν + a

)u

ν + 1

−(ν+m+1)/2

, u ≥ 0,

7

respectively. In this case, we obtain in (10) a univariate SUT distribution with ν degrees of

freedom, denoted by X ∼ SUT1,m (θ, ν), with pdf

gSUT1,m (x;θ, ν)

=

t1(x; ξ, ω, ν)Tm

(η + λ (x−ξ)

ω;ν+

(x−ξ)2ω

ν+1

(Γ− 1

ωλλT

), ν + 1

)Tm (η; Γ, ν)

, x ∈ R. (17)

We can derive an explicit expression for the mean of X ∼ SUT1,m (θ, ν), and the following

lemma is needed for this purpose.

Lemma 3 If V ∼ χ2ν/ν, where χ2

ν denotes a chi-square random variable with ν degrees of

freedom, then for any a ∈ R, b ∈ Rk and a positive definite k × k matrix ∆, and ν > 1, we

have

E[V −1/2φ

(aV 1/2

)Φk

(bV 1/2; ∆

)]=

Γ(ν−1

2

)(ν)ν/2

2√πΓ(ν2

) (ν + a2

)−(ν−1)/2Tk

(√ν − 1

ν + a2b; ∆, ν − 1

). (18)

Proof: Since V ∼ χ2ν/ν ≡ G

(ν2, ν

2

)(gamma distribution with parameters ν/2 and ν/2), we

have

E[V −1/2φ

(aV 1/2

)Φk

(bV 1/2; ∆

)]=

(ν2

) ν2

Γ(ν2

)√2π

∫ +∞

0

xν−12−1e−

12(ν+a2)xΦk

(bx1/2; ∆

)dx.

Now, by changing the variable to y = ν+a2

ν+1x and then performing some simple calculations, we

obtain

E[V −1/2φ

(aV 1/2

)Φk

(bV 1/2; ∆

)]=

Γ(ν−1

2

)(ν)ν/2

2√πΓ(ν2

) (ν + a2

)−(ν−1)/2

×E

(Φk

(√ν − 1

ν + a2bV ∗

12 ; ∆

)),

where V ∗ ∼ G(ν−1

2, ν−1

2

). Thus, the result follows from the fact E

(Φk

(√ν−1ν+a2

bV ∗12 ; ∆

))=

Tk

(√ν−1ν+a2

b; ∆, ν − 1)

.

Theorem 2 If X ∼ SUT1,m (θ, ν) , then for ν > 1,

E(X) = ξ +Γ(ν−1

2

)(ν)ν/2

2√πΓ(ν2

)Tm (η; Γ, ν)

m∑i=1

λi√γii

(ν +

η2i

γii

)−(ν−1)/2

×Tm−1

√ν − 1√ν +

η2iγii

(η −i −

ηiγiiγ−ii

); Γ−i|i, ν − 1

. (19)

Proof: The result follows easily by using Lemma 3 and the mean of the univariate SUN

distribution presented in Theorem 1.

8

3 Distributions of order statistics from a multivariate

SUE distribution

In this section, we show that in general the cdf’s of the order statistics from the SUE distribution

in (3) are mixtures of cdf’s of the univariate SUE discussed in the preceding section.

To be specific, let Xd= V | (U > 0) ∼ SUEn,m

(θ, h(m+n)

), where as before θ = (ξ,η,Ω,Γ,Λ),

and let X(n) = (X(1), · · · , X(n))T be the vector of order statistics obtained from X. Further-

more, let F(r)(t;θ, h(m+n)) and f(r)(t;θ, h

(m+n)) denote the cdf and pdf of X(r), respectively, for

r = 1, · · · , n. Also, for i = 1, · · · , n, let us partition V, ξ, Ω and Λ as follows:(V−iVi

),

(ξ−iξi

),

(Ω−i−i ω−ii

ωii

),

(ΛT−iλTi

),

so that U

V−iVi

∼ ECm+n

η

ξ−iξi

,

Γ Λ−i λiΩ−i−i ω−ii

ωii

, h(m+n)

.

In the following theorem, we first present the mixture representation for F(n)(t;θ, h(m+n)),

the cdf of X(n).

Theorem 3 The cdf of X(n) is the mixture, for t ∈ R,

F(n)(t;θ, h(m+n)) =

n∑i=1

πiGSUE1,m+n−1

(t;θ∗i , h

(m+n)), (20)

where

πi = FSUEn−1,m

(0;ξ−i−1n−1ξi,η,Ω−i−i + 1n−11

Tn−1ωii − 1n−1ω

T−ii − ω−ii1Tn−1,Γ,

ΛT−i − 1n−1λ

Ti , h

(m+n−1))

and GSUE1,m+n−1(·;θ∗i , h(m+n)) is the cdf of the univariate SUE1,m+n−1(θ∗i , h(m+n)) distribution,

with vector of parameters θ∗i = (ξi,η∗i , ωii,Γ

∗i ,λ∗i ), where, for i = 1, · · · , n,

η∗i =

(η

1n−1ξi − ξ−i

),

Γ∗i =

(Γ λi1

Tn−1 −Λ−i

Ω−i−i + 1n−11Tn−1ωii − 1n−1ω

T−ii − ω−ii1Tn−1

), (21)

λ∗i =

(λi

1n−1ωii − ω−ii

).

9

Proof: We have

F(n)(t;θ, h(m+n)) = P

(X(n) ≤ t

)= P

(V(n) ≤ t | U >0

)=

n∑i=1

P (V(n) = Vi | U >0)P (Vi ≤ t | U >0,V(n) = Vi)

=n∑i=1

P (1n−1Vi −V−i > 0 | U >0)

×P(Vi ≤ t | U > 0,1n−1Vi −V−i > 0

). (22)

Now, for i = 1, · · · , n, we have U

1n−1Vi −V−iVi

∼ ECm+n

((η∗iξi

),

(Γ∗i λ∗i

ωii

), h(m+n)

). (23)

Consequently, by using the definition of the univariate SUE distribution, we have for the i-th

term on the RHS of (22) that

P (Vi ≤ t | U > 0,1n−1Vi −V−i > 0) = GSUE1,m+n−1

(t;θ∗i , h

(m+n))

and

P (1n−1Vi −V−i > 0 | U >0) = πi,

which completes the proof of the theorem.

Corollary 2 Upon differentiating the expression of the cdf of X(n) in (20), we immediately

obtain the pdf of X(n) as

f(n)(t;θ, h(m+n)) =

n∑i=1

πigSUE1,m+n−1

(t;θ∗i , h

(m+n)),

where gSUE1,m+n−1(·;θ∗i , h(m+n)) denotes the pdf of SUE1,m+n−1(θ∗i , h(m+n)) .

A general result for the distribution of X(n) for dependent random variables is given in

Arellano-Valle and Genton (2008) wherein they focused on the case of elliptically contoured

distributions. Thus, the results presented here in Theorem 4 and Corollary 2 generalize the

corresponding ones obtained by Arellano-Valle and Genton (2008) for the case when the sample

random vector X ∼ ECn(ξ,Ω, h(n)), which is just the case of η = 0 and Λ = 0 in our results.

See also Remark 3 given below.

In a similar manner, for deriving a mixture representation for F(r)(t;θ, h(m+n)),we introduce

the following notation. Let 1 ≤ r ≤ n be an integer, and for integers 1 ≤ j1 < · · · < jr−1 ≤ n,

let Sj1···jr−1 = diag (s1, · · · , sn−1) be a (n− 1)× (n− 1) diagonal matrix such that if jr−1 = n,

then

si =

1, for i = j1, · · · , jr−2 and i = n− 1

−1, otherwise,

10

and otherwise

si =

1, for i = j1, · · · , jr−1

−1, otherwise.

In particular, Sj1···jn−1 = In−1 and Sj0 = −In−1. Furthermore, let Vj1···jr−1 = (Vj1 , · · · , Vjr−1)T

, and for i = 1, · · · , n, let the vector V−i−j1−···−jr−1 , jk 6= i, k = 1, · · · , r− 1, be obtained from

V by deleting the components Vi, Vj1 , · · · , Vjr−1 .

Theorem 4 The cdf of X(r), for r = 1, 2, · · · , n, is the mixture

F(r)(t;θ, h(m+n)) =

n∑i=1

∑j1<j2<···<jr−11≤jk≤n,jk 6=i

πij1···jr−1GSUE1,m+n−1

(t;θij1···jr−1 , h

(m+n)), (24)

where

πij1···jr−1 = GSUEn−1,m

(0;Sj1···jm

(ξ−i − 1n−1ξi

),Sj1j2···jr−1

(Ω−i−i + 1n−1

(1Tn−1ωii − ωT−ii

)−ω−ii1Tn−1

)Sj1j2···jr−1 ,Γ,Sj1···jm

(ΛT−i − 1n−1λ

Ti

), h(m+n−1)

)and

θij1···jr−1 =(ξi,ηij1···jr−1

, ωii,Γij1···jr−1 ,λij1···jr−1

), (25)

with

ηij1···jr−1=

(η

Sj1j2···jr−1

(1n−1ξi − ξ−i

) ) ,Γij1···jr−1 =

(Γ

(λi1

Tn−1 −Λ−i

)Sj1j2···jr−1

Sj1j2···jr−1

(Ω−i−i + 1n−1(1Tn−1ωii − ωT−ii)− ω−ii1Tn−1

)Sj1j2···jr−1

),

λij1···jr−1 =

(λi

Sj1j2···jr−1 (1n−1ωii − ω−ii)

).

Proof : First of all, we can write

F(r)(t;θ, h(m+n)) = P (X(r) ≤ t) = P (V(r) ≤ t | U > 0)

=n∑i=1

P (V i ≤ t, Vi = V(r) |U > 0),

where for the ith term in this last expression we have

P (Vi ≤ t, Vi = V(r) |U > 0)

=∑

j1<j2<···<jr−11≤jk≤n,jk 6=i

P(Vi ≤ t,max Vj1j2···jr−1 < Vi < min

V−i−j1−j2···−jr−1

|U > 0

),

with 1 ≤ j1 < j2 < · · · < jr−1 ≤ n and jk 6= i, for k = 1, · · · , r − 1. Then,

P(Vi ≤ t,maxVj1j2···jr−1 < Vi < min V−i−j1−j2−···−jr−1|U > 0

)= P

(max Vj1j2···jr−1 < Vi < minV−i−j1−j2−···−jr−1|U > 0

)×P

(Vi ≤ t | U > 0,maxVj1j2···jr−1 < Vi < minV−i−j1−j2−···−jr−1

).

11

Now, for i = 1, · · · , n, we have

P(maxVj1j2···jr−1 < Vi < minV−i−j1−j2−···−jr−1| U > 0

)= P

(Sj1j2···jr−11n−1Vi − Sj1j2···jr−1V−i > 0 | U > 0

)= πij1···jr−1 .

Finally, from the definition of the SUE distribution we have

P(Vi ≤ t | U > 0,maxVj1j2···jr−1 < Vi < minV−i−j1−j2−···−jr−1

)= P

(Vi ≤ t | U > 0,Sj1j2···jr−11n−1Vi − Sj1j2···jr−1V−i > 0

)= GSUE1,m+n−1

(t;θij1···jr−1 , h

(m+n)),

which completes the proof.

Corollary 3 The pdf of X(r) is obtained readily from (24) as

f(r)(t;θ, h(m+n)) =

n∑i=1

∑j1<j2<···<jr−11≤jk≤n,jk 6=i

πij1···jr−1gSUE1,m+n−1

(t;θij1···jr−1 , h

(m+n)).

Remark 1 In the special case when r = n, the result in Theorem 4 simply reduces to the result

in Theorem 3. If we take r = 1 in (24), we easily obtain the cdf of the smallest order statistic

as

F(1)(t;θ, h(m+n)) =

n∑i=1

π∗iGSUE1,m+n−1

(t;θ∗∗i , h

(m+n)),

where, for i = 1, · · · , n,

π∗i = GSUEn−1,m

(0; 1n−1ξi − ξ−i,η,Ω−i−i + 1n−1(1Tn−1ωii − ωT−ii)− ω−ii1Tn−1, Γ,

1n−1ΛTii−ΛT

−ii, h(m+n−1)

)and θ∗∗i = (ξi,η

∗∗i , ωii,Γ

∗∗i ,λ

∗∗i ), with

η∗∗i =

(η

ξ−i − ξi1n−1

),

Γ∗∗i =

(Γ Λ−i − λi1Tn−1

Ω−i−i + 1n−1(1Tn−1ωii − ωT−ii)− ω−ii1Tn−1

),

λ∗∗i =

(λi

ω−ii − 1n−1ωii

).

Remark 2 From the mixture distributions in Theorem 4, we can derive the characteristic

function of X(r). Specifically, if ϕSUE1,m+n−1(s;θ, h(m+1)) denotes the characteristic function of

12

X∼SUE1,m+n−1

(θ, h(m+1)

), we then have the characteristic function of X(r) as

Ψ(r)

(s;θ, h(m+n)

)=

n∑i=1

∑j1<j2<···<jr−11≤jk≤n,jk 6=i

πij1j2···jr−1ϕSUE1,m+n−1(s;θij1···jr−1 , h(m+n)). (26)

Remark 3 In the special case when Λ = 0 and η = 0, i.e., X ∼ ECn(ξ,Ω, h(n)), we obtain

the cdf of the rth order statistic from a multivariate elliptical distribution as follows:

F(r)(t; ξ,Ω, h(n)) =

n∑i=1

π∗ij1j2···jr−1GSUE1,n−1

(t;θ∗ij1···jr−1

, h(n)), (27)

where

π∗ij1j2···jr−1= FECn−1

(Sj1j2···jr−1

(1n−1ξi − ξ−i

);

Sj1j2···jr−1

(Ω−i−i + 1n−1(1Tn−1ωii − ωT−ii)− ω−ii1Tn−1

)Sj1j2···jr−1 , h

(n−1))

and

θ∗ij1···jr−1=(ξi,η

∗ij1···jr−1

, ωii,Γ∗ij1···jr−1

,λ∗ij1···jr−1

), (28)

with

η∗ij1···jr−1= Sj1j2···jr−1

(1n−1ξi − ξ−i

),

Γ∗ij1···jr−1= Sj1j2···jr−1

(Ω−i−i + 1n−1(1Tn−1ωii − ωT−ii)− ω−ii1Tn−1

)Sj1j2···jr−1 ,

λ∗ij1···jr−1= Sj1j2···jr−1 (1n−1ωii − ω−ii) .

The result in (27) was derived earlier by Jamalizadeh and Balakrishnan (2010) by using, how-

ever, a different but equivalent parametrization of the density in (10); see also Arellano-Valle

and Genton (2008).

3.1 Special case of SUN

For the special case of SUN, we deduce from the general expression in Theorem 4 the following

corollary.

Corollary 4 If X ∼SUNn,m (θ), the cdf and pdf of X(r) are the mixtures (for t ∈ R)

F(r)(t;θ) =n∑i=1

∑j1<j2<···<jr−11≤jk≤n,jk 6=i

πij1···jr−1GSUN1,m+n−1

(t;θij1···jr−1

)

and

f(r)(t;θ) =n∑i=1

∑j1<j2<···<jr−11≤jk≤n,jk 6=i

πij1···jr−1gSUN1,m+n−1

(t;θij1···jr−1

),

13

respectively, where the mixing probabilities πij1···jr−1 are

πij1···jr−1 = GSUNn−1,m

(0;Sj1j2···jr−1

(ξ−i − 1n−1ξi

),Sj1j2···jr−1

(Ω−i−i + 1n−1(1Tn−1ωii − ωT−ii)

−ω−ii1Tn−1

)Sj1j2···jr−1 ,Γ,Sj1j2···jr−1

(ΛT−i − 1n−1λ

Ti

)),

andGSUN1,m+n−1

(· ;θij1···jr−1

)and gSUN1,m+n−1

(· ;θij1···jr−1

)denote the cdf and pdf, respectively,

of the univariate SUN1,m+n−1

(θij1···jr−1

)distribution, with θij1···jr−1 being as given in (25).

Remark 4 In the above case of SUN, the MGF of X(r) is

M(r) (s;θ) =n∑i=1

∑j1<j2<···<jr−11≤jk≤n,jk 6=i

πij1···jr−1MSUN1,m+n−1

(s;θij1···jr−1

), (29)

where MSUN1,m+n−1

(·;θij1···jr−1

)is the MGF of SUN1,m+n−1

(θij1···jr−1

).

Corollary 5 If in the SUN case, in addition we have Λ = 0, i.e., X ∼ Nn(ξ,Ω), then we

obtain the cdf of the rth order statistic from multivariate normal distribution as

F(r)(t; ξ,Ω) =n∑i=1

π∗ij1j2···jr−1GSUN1,n−1

(t;n− 1,θ∗ij1···jr−1

),

where

π∗ij1j2···jr−1= Φn−1

(Sj1j2···jr−1

(1n−1ξi − ξ−i

);

Sj1j2···jr−1

(Ω−i−i + 1n−1(1Tn−1ωii − ωT−ii)− ω−ii1Tn−1

)Sj1j2···jr−1

)and θ∗ij1···jr−1

is as in (28).

This last result was derived earlier by Jamalizadeh and Balakrishnan (2010) by using a

different parametrization of the density in (12). For η = 0 and Λ = 0, we obtain the distribution

of X(r) when the sample random vector X = (X1, . . . , Xn)T ∼ Nn(ξ,Ω), which was also studied

by Arellano-Valle and Genton (2007, 2008).

Additional byproducts of the above results are related with the extension of the multivariate

log-normal distribution to the multivariate log-unified skew-normal distribution. In this regard,

an n-dimensional random vector W = (W1, · · · ,Wn)T is said to have a multivariate log-unified

skew-normal distribution with parameter θ = (ξ,η,Ω,Γ,Λ), denoted by W ∼LSUNn,m (θ),

if

log W = (logW1, · · · , logWn)Td= X ∼SUNn,m (θ) .

Recently, Marchenko and Genton (2010) discussed a special case of this distribution with ap-

plication to precipitation data.

If W(n)=(W(1), · · · ,W(n)

)Tdenotes the vector of order statistics from W = (W1, · · · ,Wn)T ,

we show in the following corollary that the moments of W(r), for r = 1, · · · , n, can be readily

obtained from the MGF of X(r) in (28).

14

Corollary 6 We have, for s ∈ R,

E(W s

(r)

)= M(r) (s;θ) .

Proof : The proof follows readily from the fact that

logW(r) = X(r), for r = 1, · · · , n,

where X(r) is the rth order statistic from X = (X1, · · · , Xn)T ∼ SUNn,m (θ). It is useful to mention here that the corresponding result for the multivariate log-normal

distribution has been given earlier by Jamalizadeh and Balakrishnan (2010).

3.2 Special case of SUT

For the special case of SUT, we deduce from the general expression in Theorem 4 the following

corollary.

Corollary 7 If X ∼SUTn,m (θ, ν), then the cdf and pdf of X(r) are the mixtures (for t ∈ R)

F(r)(t;θ, ν) =n∑i=1

∑j1<···<jr−1

1≤jk≤n,jk 6=i

πij1···jr−1GSUT1,m+n−1

(t;θij1···jr−1 , ν

)and

f(r)(t;θ, ν) =n∑i=1

∑j1<···<jr−1

1≤jk≤n,jk 6=i

πij1···jr−1gSUT1,m+n−1

(t;θij1···jr−1 , ν

),

respectively, where

πij1···jr−1 = GSUTn−1,m

(0;Sj1···jr−1

(ξ−i − 1n−1ξi

),Sj1···jr−1

(Ω−i−i + 1n−1(1Tn−1ωii − ωT−ii)

−ω−ii1Tn−1

)Sj1···jr−1 ,Γ,Sj1···jr−1

(ΛT−i − 1n−1λ

Ti

), ν),

and GSUT1,m+n−1

(· ;θij1···jr−1 , ν

)and gSUT1,m+n−1

(· ;θij1···jr−1 , ν

)are, respectively, the cdf and

pdf of the univariate SUT1,m+n−1

(· ; ,θij1···jr−1 , ν

)distribution, with θij1···jr−1 being as in (25).

Corollary 8 If in the SUT case, in addition we have Λ = 0 and η = 0, i.e., X ∼ tn(ξ,Ω, ν),

then we obtain the cdf of the rth order statistic from multivariate t distribution as (for t ∈ R)

F(r)(t; ξ,Ω, ν) =n∑i=1

π∗ij1···jr−1GSUT1,n−1

(t;θ∗ij1···jr−1

, ν),

where

π∗ij1···jr−1= Tn−1

(Sj1···jr−1

(1n−1ξi − ξ−i

);

Sj1···jr−1

(Ω−i−i + 1n−1(1Tn−1ωii − ωT−ii)− ω−ii1Tn−1

)Sj1···jr−1 , ν

)and θ∗ij1···jr−1

is as in (28). This result has been derived by Jamalizadeh and Balakrishnan (2010)

by using a different parametrization of the density in (17).

15

4 Distribution of linear combinations of order statistics

from a multivariate SUE distribution

In this section, we consider distribution of linear combinations of the order statistics from a

SUE distribution. We show that this is, in fact, a mixture of multivariate SUE distributions. As

in the last section, we assume that Xd= V | (U > 0) ∼SUEn,m

(θ, h(m+n)

), where θ is as before

and X(n) = (X(1), · · · , X(n))T is the vector of order statistics obtained from X. More specifically,

if L ∈ Rp×n is a matrix of rank p and, we are interested in the distribution of the linear combina-

tions LX(n). For this purpose, we denote by V(n) = (V(1), · · · , V(n))T the vector of order statistics

obtained from V, and then note that V(n) ∈ P (V), where P (V) = Vi = PiV; i = 1, · · · , n!is the collection of random vectors Vi corresponding to the n! different permutations of the com-

ponents of V, and Pi ∈ Rn×n are the permutation matrices with Pi 6= Pj for all i 6= j. Further,

let D ∈ R(n−1)×n be a difference matrix such that DV = (V2 − V1, V3 − V2, · · · , Vn − Vn−1),

i.e., the i-th row of D is eTn,i+1−eTn,i, i = 1, · · · , n−1, where en,1, · · · , en,n are the n-dimensional

unit basis vectors, and ξi = Piξ,Λi = Pi Λ and Ωi = PiΩPTi .

Theorem 5 The cdf of LX(n) is the mixture

FLX(n)

(t;θ, h(m+n)

)=

n!∑i=1

ωiGSUEp,m+n−1

(t;θi, ϕ

(m+n+p−1)), t ∈Rp, (30)

where for i = 1, · · · , n!,

ωi = GSUEn−1,m

(0;−Dξi,η,DΩiD

T ,Γ,−DΛi, ϕ(m+n−1)

)and GSUEp,m+n−1

(·;θi, ϕ(m+n+p−1)

)is the cdf of SUEp,m+n−1

(θi, ϕ

(m+n+p−1))

with parameters

θi =

(Lξi,

(η

Dξi

),LΩiL

T ,

(Γ ΛT

i DT

DΩiDT

),(LΛi,LΩiD

T))

. (31)

Proof : We have

FLX(n)

(t;θ, h(m+n)

)= P

(LX(n) ≤ t

)= P

(LV(n) ≤ t | U > 0

)=

n!∑i=1

P(LVi ≤ t,V(n) = Vi | U > 0

)=

n!∑i=1

P (LVi ≤ t,DVi ≥ 0 | U > 0)

=n!∑i=1

P (DVi ≥ 0 | U > 0)P (LVi ≤ t | U > 0,DVi ≥ 0) . (32)

Now, for i = 1, · · · , n!, U

DVi

LVi

∼ ECm+n+p−1

η

DξiLξi

,

Γ ΛTi DT ΛT

i LT

DΩiDT DΩiL

T

LΩiLT

, ϕ(m+n+p−1)

.

16

Consider the ith term in (32). From the definition of the SUE distribution, we have

P (LVi ≤ t | U > 0,DVi > 0) = GSUEp,m+n−1

(t;θi, ϕ

(m+n+p−1))

and

P (DVi ≥ 0 | U > 0) = ωi,

which completes the proof.

Remark 5 Upon differentiating the expression of the cdf of LX(n) in (30), we readily obtain

the density function of LX(n) as

fLX(n)

(t;θ, h(m+n)

)=

n!∑i=1

ωigSUEp,m+n−1

(t;θi, ϕ

(m+n+p−1)), t ∈Rp.

As a byproduct of Theorem 5, we obtain the following result which shows that the distri-

bution of a linear combination of order statistics is a mixture of the univariate unified skew-

elliptical distributions.

Corollary 9 The cdf of aTX(n), where a = (a1, · · · , an)T ∈ Rn, is the mixture

F aTX(n)

(t;θ, h(m+n)

)=

n!∑i=1

ωiGSUE1,m+n

(t;θi, ϕ

(m+n)),

where

ωi = GSUEn−1,m

(0; ,η,DΩiD

T ,Γ,−DΛi, ϕ(m+n−1)

)and

θi =

(aTξi,

(η

Dξi

), aTΩia,

(Γ ΛT

i DT

DΩiDT

),

(ΛTi a

DΩia

)).

It is important to mention here that in the special case when∑n

i=1 ai 6= 0, the cdf of aTX(n)

is in fact a mixture of cdf’s of the univariate SUE distributions in (10). Consequently, in

Corollary 9 ,the characteristic generator function ϕ(n) can be replaced by the respective density

generator function h(n).

Remark 6 In the special case when Λ = 0 and η = 0, i.e., X ∼ ECn(ξ,Ω, h(n)), we obtain

the cdf of linear combinations of order statistics from an elliptical distribution as

FLX(n)

(t;ξ,Ω, h(n)

)=

n!∑i=1

ω∗iGSUEp,n−1

(t;θ∗i , ϕ

(n+p−1)), t ∈Rp, (33)

where ω∗i = GECn−1

(Dξi; DΩiD

T , ϕ(n−1))

and θ∗i =(Lξi,Dξi,LΩiL

T ,DΩiDT ,LΩiD

T). This

result has been presented earlier by Jamalizadeh and Balakrishnan (2010) using a reparametrized

form of the density in (3); see also Arellano-Valle and Genton (2008).

Let L =(LT

1 ,LT2

)T ∈ Rp×n be a matrix of rank p, where L1 ∈ Rp1×n and L2 ∈ Rp2×n are

matrices of rank p1 and p2 = p − p1, respectively. Then, the following corollary presents the

17

conditional distribution of L2X(n) | (L1X(n) = t1). The proof of this result follows directly from

Theorem 5 and Lemma 1.

Corollary 10 We have, for t1 ∈ Rp1 ,

FL2X(n)|(L1X(n)=t1)

(t2;θ, h(m+n)

)=

n!∑i=1

ωiGSUEp2,m+n−1

(t2;θ2.1

i , ϕ(m+n+p2−1)qi(t1)

), t2∈Rp2 , (34)

where

θ2.1i =

(ξ2.1i (t1),η2.1

i (t1),Ω22.1i ,Γ2.1

i , Λ2.1i

)(35)

with

ξ2.1i (t1) = L2ξi + L2ΩiL

T1

(L1ΩiL

T1

)−1(t1 − L1ξi) ,

η2.1i (t1) =

(η

Dξi

)+

(ΛTi LT

1

DΩiLT1

)(L1ΩiL

T1

)−1(t1 − L1ξi) ,

Ω22.1i = L2ΩiL

T2 − L2ΩiL

T1

(L1ΩiL

T1

)−1L1ΩiL

T2 ,

Γ2.1i =

(Γ ΛT

i DT

DΩiDT

)−(

ΛTi LT

1

DΩiLT1

)(L1ΩiL

T1

)−1 (L1Λi,L1ΩiD

T),

Λ2.1i =

(L2Λi,L2ΩiD

T)− L2ΩiL

T1

(L1ΩiL

T1

)−1 (L1Λi,L1ΩiD

T),

qi(t1) = (t1 − L1ξi)T (L1ΩiL

T1

)−1(t1 − L1ξi) .

4.1 Special case of multivariate SUN

When X ∼SUNn,m (θ), the cdf of LX(n) can be easily deduced from the general form in (30).

Corollary 11 If X ∼SUNn,m (θ) , then the cdf of LX(n) is the mixture

FLX(n)(t;θ) =

n!∑i=1

ωiGSUNp,m+n−1 (t;θi) , t ∈Rp,

where θi is as given in Theorem 5, and for i = 1, · · · , n!,

ωi = GSUNn−1,m

(0;−Dξi,η,DΩiD

T ,Γ,−DΛi

).

In this case, the MGF of LX(n) can also be easily obtained from the mixture form in

Corollary 11 by using (4).

Remark 7 The MGF of LX(n) is

MLX(n)(s;θ) =

n!∑i=1

ωiMSUNp,m+n−1 (s;θi) , s ∈ Rp, (36)

where MSUNp,m+n−1 (· ;θi) is the MGF of SUNp,m+n−1 (θi).

18

Next, when X ∼SUNn,m (θ), the cdf of aTX(n) can be easily deduced from the general form

in Corollary 9 as given in the following corollary.

Corollary 12 If X ∼SUNn,m (θ), then the cdf of aTX(n) is the mixture

FaTX(n)(t;θ) =

n!∑i=1

ωiGSUN1,m+n (t;θi) ,

where in this case, for i = 1, · · · , n!,

ωi = GSUNn−1,m

(−Dξi; 0,η,DΩiD

T ,Γ,−DΛi

)and θi is as given in Corollary 9. This result has been presented earlier by Jamalizadeh and

Balakrishnan (2010).

From the MGF of LX(n) given in Remark 7, we can readily derive the MGF of aTX(n) in

the SUN case, which is presented in the following corollary.

Corollary 13 If X ∼SUNn,m (θ), the MGF of aTX(n) is given by

M aTX(n)(s;θ) =

n!∑i=1

ωiMSUN1,m+n (s;θi) , (37)

where θi is as given in Corollary 9.

Corollary 14 If W = (W1, · · · ,Wn)T ∼LSUNn,m (θ) and W(n) =(W(1), · · · ,W(n)

)Tare the

corresponding order statistics, then for a = (a1, · · · , an)T ∈ Rn, we have

E(W a1

(1) · · ·Wan(n)

)= MaTX(n)

(1;θ) .

4.2 Special case of multivariate SUT

When X ∼SUTn,m (θ, ν), the cdf of LX(n) can be easily deduced from the general form in (30).

Corollary 15 If X ∼SUTn,m (θ, ν), then the cdf of LX(n) is the mixture

FLX(n)(t;θ, ν) =

n!∑i=1

ωiGSUTp,m+n−1 (t;θi, ν) , t ∈Rp,

where

ωi = GSUTn−1,m

(−Dξi; 0,η,DΩiD

T ,Γ,−DΛi, ν)

and θi is as given in Theorem 5.

In this case, by using the mixture forms in Corollaries 10 and 1, we have the following result.

19

Corollary 16 We have

FL2X(n)|(L1X(n)=t1) (t2;θ, ν) =n!∑i=1

ω∗iGSUTp2,m+n−1

(t2;θ∗2.1i , ν + p1

), t1∈Rp1 , t2∈Rp2 ,

where

θ∗2.1i =

(ξ2.1i (t1),η2.1

i (t1),ν + q1(t1)

ν + p1

Ω22.1i ,

ν + q1(t1)

ν + p1

Γ2.1i ,

ν + q1(t1)

ν + p1

Λ2.1i

)and ξ2.1

i (t1),η2.1i (t1),Ω22.1

i ,Γ2.1i ,Λ2.1

i are as in Corollary 10.

5 Concluding remarks

Crocetta and Loperfido (2005) derived the exact joint distribution of linear combinations of

order statistics from skew-normal random variables (i.i.d. case). Subsequently, Jamalizadeh and

Balakrishnan (2008) derived the exact distribution of order statistics from bivariate skew-normal

and bivariate skew-t distributions. In this paper, we have extended these results by deriving the

exact joint distribution of linear combinations of order statistics from the multivariate unified

skew-elliptical distributions in (3).

Arellano-Valle et al. (2006) defined a selection distribution as the conditional distribution

of V, given U ∈ C. Specifically, a n-dimensional random vector X is said to have a multi-

variate selection distribution, denoted by X ∼ SLCTn,m, with parameters depending on the

characteristics of U,V and C, if

Xd= V | (U ∈ C) .

In the special case when V and U have jointly an elliptical distribution as in (1), then X

is said to have a multivariate selection elliptical distribution, denoted by X ∼ SLCT −ECn,m

(θ, h(m+n), C

), and has its density function as

gSLCT−ECn,m

(x;θ, h(m+n), C

)=fECn

(x; ξ,Ω, h(n)

)FECm

(C;η + ΛTΩ−1 (x− ξ) ; Γ−ΛTΩ−1Λ, h

(m)w(x)

)FECm (C;η; Γ, h(m))

,

(38)

where FECr

(C;µY; ΣY, h

(r))

denotes P (Y ∈C) when Y ∼ ECr(µY,ΣY, h

(r)). Here, we

specifically have taken C as

C = u ∈ Rm | u > 0

and then discussed the distributions of order statistics and linear combinations of order statistics

from the distribution in (3). Recently, Jamalizadeh et al. (2010) studied the exact distributions

of order statistics and linear combination of order statistics from a bivariate selection normal

20

distribution by taking C = u ∈ R | a ≤ u ≤ b. It will, of course, be of interest to study

further the distributions of order statistics and linear combinations of order statistics from (38)

for a general C. We are currently working on this problem and hope to report these findings

in a future paper.

Acknowledgements

The authors thank an anonymous reviewer and the editor for making some useful suggestions

on an earlier version of this manuscript which led to this improved version. Arellano-Valles

research was partially supported by grant FONDECYT 1120121-Chile while N. Balakrishnan’s

research was supported by the Natural Sciences and Engineering Research Council of Canada.

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