1_xxx_gupta.dvi


EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS

Vol. 3, No. 2, 2010, 128-140

ISSN 1307-5543 – www.ejpam.com

A Matrix Variate Skew Distribution

Deniz Akdemir∗1, Arjun K. Gupta2

1 Department of Mathematics and Statistics, Bowling Green State University, Bowling Green, Ohio

43403, USA (Now at the Ohio Northern University, Ada, Ohio, 45810, USA)
2 Department of Mathematics and Statistics, Bowling Green State University, Bowling Green, Ohio

43403, USA

Abstract. Typical multivariate analysis assumes independence among the individual observations as

well as elliptical symmetry of distributions. In many situations these assumptions may be too restric-

tive. This paper studies a class of flexible matrix variate distribution models that can represent both

skewed and symmetric distributions which can also account for dependence among individual obser-

vations. We derive the moment generating function and study linear and quadratic forms of interest

that help understand the properties of these models.

2000 Mathematics Subject Classifications: Primary 62H10, Secondary 62H05.

Key Words and Phrases: Skew Normal Distribution, Normal Distribution, Multivariate Distribution,

Matrix Variate Skew Normal Distribution.

1. Introduction

1.1. Distribution of Random Sample

In order to make inferences about the population parameters of a k dimensional paramet-

ric distribution, we work with a random sample of n individuals from this population which

can be represented by a k× n matrix X . Typical multivariate analysis assumes independence

among the individuals. In many situations this assumption may be too restrictive. For exam-

ple, many data collection and sample designs involve some overlapping between interviewer

workload and the sampling units (clusters). A proportion of the measurement variance which

is due to interviewers is reflected to some degree in the sampling variance calculations. In

the literature, the variable effects that interviewers have on respondent answers are some-

times labeled the correlated response variance [3]. Matrix variate distributions can be used to

account for the dependence among individual vector observations.

∗Corresponding author.

Email addresses: d-akdemir�onu.edu (D. Akdemir), gupta�bgsu.edu (A.K. Gupta)

http://www.ejpam.com 128 c© 2010 EJPAM All rights reserved.



D. Akdemir, A.K. Gupta / Eur. J. Pure Appl. Math, 3 (2010), 128-140 129

Unarguably, the most commonly used family of distributions is the normal family. Under

normality assumption, the matrix random variable X will have the following density [6]:

φk×n(X ; M ,AA′, B′B) =
etr(−1

2
(AA′)−1(X −M)(B′B)−1(X −M)′)

(2π)nk/2|AA′|n/2|B′B|k/2 (1)

where A is a k×k matrix, B is an n×n matrix such that AA′ and B′B are positive definite; M is a

k×n matrix. Like the multivariate case, the matrix A determines how the variables are related,

also the matrix B is introduced to account for dependence among individual observations. In

this family, orthogonality of rows of the matrix A or the columns of the matrix B is equivalent

to independence of rows or columns of the random matrix X .

In the remainder of this paper we will use φ(.) and Φ(.) for the density and the cdf of the

normal random variables. When we want to refer to the multivariate or matrix variate forms

of these functions, we will use subindices to describe the dimensions. For example, we will

write Φk×n(X ; M ,A, B) to represent the cdf of a matrix variate normal random variable with

parameters M , A and B evaluated at X .

1.2. Multivariate Skew Distributions

A k dimensional random vector x with pdf f (.) is centrally symmetric about 0 if f (t ) =

f (−t ) for all t in the domain of x . In this section, we study a family of multivariate skew

symmetric densities generated by centrally symmetric densities.

Theorem 1. Let g(.) be the k-dimensional jpdf for k independent variables centrally symmetric

about 0, H(.) be an absolutely continuous cumulative distribution function with H ′(.) symmetric

about 0, α = (α1,α2, . . . ,αk)
′ be a k−dimensional real vector, and e j for j = 1,2, . . . , k are the

elementary vectors of Rk. Then

f (y,α) = 2k g(y)

k
∏

j=1

H(α je
′
j y) (2)

defines a probability density function of y .

Proof. First note that f (y)≥ 0 for all y ∈ Rk. We need to show that

k(α1,α2, ...,αk) =

∫

Rk

2k g(y)

k
∏

j=1

H(α je
′
j y)dy= 1.

Observe that,

∂

∂ αℓ
k(α1,α2, ...,αk) =

∫

Rk

d

dαℓ
2k g(y)

k
∏

j=1

H(α je
′
j y)dy

=

∫

Rk

2k yℓH
′(αℓe′ℓy)

k
∏

j 6=ℓ
H(α je

′
j y)g(y)dy



D. Akdemir, A.K. Gupta / Eur. J. Pure Appl. Math, 3 (2010), 128-140 130

= 0.

The first equality is true because of Lebesgue dominated convergence theorem; the last

equality is first due to independence by seeing

Eg(y)(yℓH
′(αℓe′ℓy)

k
∏

j 6=ℓ
H(α je

′
j y)) = Eg(y)(yℓH

′(αℓe′ℓy))Eg(y)(

k
∏

j 6=ℓ
H(α je

′
j y))

and because g(.) is centrally symmetric about 0, yℓH
′(αℓe′ℓy) is an odd function of yℓ,

Eg(y)(yℓH
′(αℓe′ℓy)) = 0.

Hence, k(α1,α2, ...,αk) is constant as a function of α j for all j = 1,2, . . . , k; and when all

αi = 0, k(α1,α2, ...,αk) = 1. This concludes the proof.

We will write z ∼ ss
g ,H

k
(z;0, Ik,α) or z ∼ ss

g ,H

k
(α) for a random variable with density

given by (2). For the random vector y = Az+µ where A is a nonsingular matrix we will write

y ∼ ss
g ,H

k
(z;µ,A,α) or y ∼ ss

g ,H

k
(µ,A,α).

In the next theorem, we relate the distribution of the even powers of a skew symmetric

random variable to those of its kernel’s.

Theorem 2. Let x be a random vector with probability density function g(x ), and y be the

random vector with probability density function

f (y,α) = 2k g(y)

k
∏

j=1

H(α je
′
j y)

where g(.), H(.) and α are defined as in Theorem 1. Then,

1. the even moments of y and x are the same, i.e E(y y ′)p = E(x x ′)p for p even and

E(y ′y)m = E(x ′x )m for any natural number m,

2. y ′y and x ′x have the same distribution.

Proof. It suffices to show that

E(x
n1

1
x

n2

2
. . . x

nk

k
) = E(y

n1

1
y

n2

2
. . . y

nk

k
)

for n1, n2, . . . , nk even.

Let Ψy(t ) be the characteristic function of y . Then,

Ψy(t ) =

∫

Rk

eit′y2k g(y)

k
∏

j=1

H(α je
′
j y)dy. (3)

Let n1 + n2 + . . . + nk = n. Taking the nth
j

partial derivatives of (3) with respect to t j for

j = 1,2, . . . , k and putting t = 0



D. Akdemir, A.K. Gupta / Eur. J. Pure Appl. Math, 3 (2010), 128-140 131

∂ nψy (t)

∂ t
n1

1
∂ t

n2

2
. . . ∂ t

nk

k
|t=0

=

∫

Rk

∂ n

∂ t
n1

1
∂ t

n2

2
. . .∂ t

nk

k

eit′y2k

×
k
∏

j=1

H(α je
′
j y)g(y)dy|t=0

=

∫

Rk

[eit′y2kin
k
∏

j=1

H(α je
′
j y)]

× [

k
∏

ℓ=1

y
nℓ
ℓ
]g(y)dy|t=0

=

∫

Rk

[2kin
k
∏

j=1

H(α je
′
j y)][

k
∏

ℓ=1

y
nℓ
ℓ
]g(y)dy. (4)

Taking derivative of (4) with respect to αm,

∂ (
∫

Rk[2
kin
∏k

j=1 H(α je
′
j y)][
∏k

ℓ=1 y
nℓ
ℓ
]g(y)dy)

∂ αm

= 2kinEg(y)[y
(nm+1)
m H ′(αme′m y)[

∏

j 6=m

y
nℓ
ℓ

H(α je
′
j y)]

= 2kinEg(y)[y
(nm+1)
m H ′(αme′m y)]Eg(y)[

∏

j 6=m

y
nℓ
ℓ

H(α je
′
j y)]

= 0

The first equality is true because of Lebesgue dominated convergence theorem, the second

equality due to the independence of components. The last equality is due to the fact that

y(nm+1)
m H ′(αm ym)

is an odd function of ym and g(.) is centrally symmetric about 0.

Therefore, for n1, n2, . . . , nk even, E(y
n1

1
y

n2

2
. . . y

nk

k
) is constant as a function of αm.

If all αm = 0 then f (x ) = g(x ) and therefore

E(x
n1

1 x
n2

2 . . . x
nk

k
) = E(y

n1

1 y
n2

2 . . . y
nk

k
)

Finally,

E(x
n1

1 x
n2

2 . . . x
nk

k
) = E(y

n1

1 y
n2

2 . . . y
nk

k
)

is true for all αm. The required results follow immediately.

A skew normal density is obtained from normal kernel in the following example.



D. Akdemir, A.K. Gupta / Eur. J. Pure Appl. Math, 3 (2010), 128-140 132

Example 1. In Theorem 1 above, let g(.) = φk(.), where φk(.) is the k-dimensional standard

normal density function. Also let H(.) and α be defined as in Theorem 1. We can construct a

density for k-dimensional joint p.d.f ’s of the form

f (y ,α) = 2kφk(y)

k
∏

j=1

H(α je
′
j y) (5)

This density will be called the generalized skew normal probability density function and will

be represented by snH
k
(y;0, I ,α). For the random vector y with this density we will write y ∼

snH
k
(0, I ,α). If H(.) is taken as Φ(.), the cdf of the standard normal variable, then we will drop

the super index H and this defines the skew normal probability density function and the skew

normal random vector.

Using Theorem 2 we can relate some properties of the snk(0, I ,α) random vector with its

kernel, the standard multivariate normal random vector with density φk(.). Let x ∼ φk(x ), and

y ∼ snk(0, I ,α). Then,

1. the even moments of y and x are the same, i.e E(y y ′)p = E(x x ′)p for p even and

E(y ′y)m = E(x ′x )m for any natural number m,

2. y ′y and x ′x both have χ2
k

distribution.

2. Matrix Variate Skew Distributions

Chen and Gupta extend the matrix normal distribution to accommodate skewness in the

following form [4]:

f1(X ;Σ,Ψ, b) = c∗1φk×n(X ;0,Σ,Ψ)Φn(X
′b;0,Ψ) (6)

where c∗1 = (Φn(0;0, (1 + b′Σb)Ψ))−1. A drawback of this definition is that it allows inde-

pendence only over its rows or columns, but not both. Harrar and Gupta [7] give two more

definitions for the matrix variate skew normal density:

f2(X ;Σ,Ψ, b,Ω) = c∗2φk×n(X ;0,Σ,Ψ)Φn(X
′b;0,Ω) (7)

and

f3(X ;Σ,Ψ, b, B) = c∗3φk×n(X ;0,Σ,Ψ)Φ(t r(B′X ), 0,1) (8)

where c∗2 = (Φn(0, (Ω + b′Σb)Ψ))−1, c∗3 = 2; Σ, Ψ, and Ω are positive definite covariance

matrices of dimensions k, n and n respectively, B is a matrix of dimension k× n. Note that if

Ω = Ψ then f2 is the same as f1. Although, more general than f1, the density f2 still does not

permit independence of rows and columns simultaneously.

A very general definition of skew symmetric variable for the matrix case can be obtained

from matrix variate selection models. Suppose X is a k× n random matrix with density f (X ),

let g(X ) be a weight function. A weighted form of density f (X ) is given by

h(X ) =
f (X )g(X )
∫

R
k×n g(X ) f (X )dX

. (9)



D. Akdemir, A.K. Gupta / Eur. J. Pure Appl. Math, 3 (2010), 128-140 133

When the sample is only a subset of the population then the associated model would be called

a selection model.

In the next section, a construction for a family of matrix variate skew-symmetric densities

that allows for independence among both variables and individuals is studied.

2.1. Matrix variate Skew Symmetric Distribution

To define a matrix variate distribution from the multivariate skew symmetric distribution

first assume that z i ∼ ss
g ,H

k
(0, Ik,αi) for i = 1,2, . . . , n are independently distributed random

variables. Write Z = (z1, z2, . . . , zn). We can write the density of X as a product as follows,

n
∏

i=1

2k g∗(Z)
k
∏

j=1

H(α jie
′
j z i).

This is equal to

2nk g∗∗(Z)
n
∏

i=1

k
∏

j=1

H(α jie
′
j Zc i).

Let A, and B be nonsingular symmetric matrices of order k, and n respectively, also assume

M is a k× n matrix. Define the matrix variate skew symmetric variable as X = AZB+M .

Definition 1. (Matrix Variate Skew-Symmetric Density) Let g(.) be a density function symmetric

about 0, H(.) be an absolutely continuous cumulative distribution function with H ′(.) symmetric

about 0. A variable X has matrix variate skew symmetric distribution if it has probability density

function

2nk g∗∗(A−1(X −M)B−1)
∏n

i=1

∏k

j=1 H(α jie
′
j(A
−1(X −M)B−1)c i)

|A|n|B|k (10)

where α ji are real scalars, M ∈ Rk×n, A and B be nonsingular symmetric matrices of order k and

n respectively. Finally, g∗∗(X ) =
∏n

i=1

∏k

j=1 g(yi j). The density is called matrix variate skew-

symmetric density with location parameter M , scale parameters (A, B), and shape parameter

∆= (α ji), and it is denoted by mss
g ,H

k×n
(M ,A, B,∆).

Let Z ∼ mss
g ,H

k×n
(0k×n, Ik, In,∆). The moment generating function of Z evaluated at Tk×n ∈

R
k×n is MZ (Tk×n) and can be obtained as follows:

MZ (Tk×n) = E(etr(T ′k×nZ))

=

∫

R
k×n

etr(T ′k×nZ)2k g∗∗(Z)
n
∏

i=1

k
∏

j=1

H(α jie
′
j Zc i)dZ

= Eg∗∗(Z)(etr(T ′k×nZ)

n
∏

i=1

k
∏

j=1

H(α jie
′
j Zc i)).



D. Akdemir, A.K. Gupta / Eur. J. Pure Appl. Math, 3 (2010), 128-140 134

Let X = AZB +M where A (k× k), B (n× n) and M (k× n) are constant matrices. Then

moment generating function of X evaluated at Tk×n is MX (Tk×n):

MX (Tk×n) = etr(T ′k×nM)MZ (A
′Tk×nB′)

= etr(T ′k×nM)Eg∗∗(Z)(etr((BT ′k×nA)Z)

n
∏

i=1

k
∏

j=1

H(α jie
′
j Zc i)).

Definition 2. (Matrix Variate Skew Symmetric Distributions) Let g(.) be a density function

symmetric about 0, H(.) be an absolutely continuous cumulative distribution function with H ′(.)
symmetric about 0. Let zi j ∼ f (zi j ,α ji) = 2g(zi j)H(α jiz) for i = 1,2, . . . , n, and j = 1,2, . . . , k

be independent variables. Then the matrix variate random variable Z = (zi j) has density

2nk g∗∗(Z)
n
∏

i=1

k
∏

j=1

H(α jie
′
j Zc i)

where g∗∗(z) =
∏n

i=1

∏k

j=1 g(zi j), and e′j and c′i are the elementary vectors of the coordinate

system Rk and Rn respectively. Let X = AZB+M where A (k× k), B (n× n) and M (k× n) are

constant matrices. Then the random variable X = AZB +M has matrix variate skew symmetric

distribution with location parameter M , scale parameters (A, B), and shape parameter∆= (α ji).

We denote this by X ∼ MSS
g ,H

k×n
(M ,A, B,∆).

2.2. Matrix Variate Skew-Normal Distribution

Definition 3. (Matrix Variate Skew Normal Density). We call

2knφk×n(A
−1(X −M)B−1)

∏k

j=1

∏n

i=1Φ(α jie
′
j(A
−1(X −M)B−1)c i)

|A|n|B|k (11)

the matrix variate skew normal density with location parameter M , scale parameters (A, B), and

shape parameter ∆. We denote it by msnk×n(M ,A, B,∆).

We will need the following lemmas: See Zacks [9] and Chen and Gupta [4].

Lemma 1. Let z ∼ φk(z). For scalar b, a ∈ Rk, and for Σ a positive definite matrix of order k

E(Φ(b+ a′Σ1/2z)) = Φ( b

(1+a′Σa)1/2
).

Lemma 2. Let Z ∼ φk×n(Z). For scalar b, a ∈ Rk, and for A and B positive definite matrices of

order k and n respectively, E(Φn(b+ a′AZB)) = Φn(a, (1+ a′AA′a)1/2B).

Let Z ∼ msnk×n(0k×n, Ik, In,∆). Then, the moment generating function of Z evaluated at

Tk×n is MZ (Tk×n). It can be obtained as follows:

MZ (Tk×n) = Eφk×n(Z)
(etr(T ′k×nZ)

n
∏

i=1

k
∏

j=1

Φ(α jie
′
j Zc i))



D. Akdemir, A.K. Gupta / Eur. J. Pure Appl. Math, 3 (2010), 128-140 135

=
2k

(2π)k/2

n
∏

i=1

∫

Rk

e−
1

2
z′

i
z i+t i

′z i

k
∏

j=1

Φ(α jie
′
j z i)dz

=
2k

(2π)k/2

n
∏

i=1

∫

Rk

e−
1

2
(z′

i
z i−2t i

′z i)
k
∏

j=1

Φ(α jie
′
j z i)dz

=

n
∏

i=1

2ke
1

2
t i
′ t i

(2π)k/2

∫

Rk

e−
1

2
(z′

i
z i−2t i

′z i+t i
′ t i)

k
∏

j=1

Φ(α jie
′
j z i)dz

=

n
∏

i=1

2ke
1

2
t i
′ t i

(2π)k/2

∫

Rk

e−
1

2
(z i−t i)

′(z i−t i)
k
∏

j=1

Φ(α jie
′
j z i)dz

=

n
∏

i=1

2ke
1

2
t i
′ t i

(2π)k/2

k
∏

j=1

∫

R

e−
1

2
(zi j−(t )i j)

2

Φ(α ji z i j)dzi j

=

n
∏

i=1

2ke
1

2
t i
′ t i

k
∏

j=1

∫

R

1
p

(2π)
e−

1

2
(yi j )

2

Φ(α ji y i j +α ji(t )i j)d yi j

=

n
∏

i=1

2ke
1

2
t i
′ t i

k
∏

j=1

Φ(
α ji(t )i j
p

(1+α ji
2)
)

= 2nketr(
1

2
Tk×n

′Tk×n)

n
∏

i=1

k
∏

j=1

Φ(
α ji(T )i j
p

(1+α ji
2)
).

Let X = AZB + M for constant (k × k) matrix A, (n× n) matrix B and k × n dimensional

constant matrix M . Then the moment generating function of X evaluated at Tk×n ∈ Rk×n is

MX (Tk×n), this can be obtained as follows:

MX (Tk×n) = 2nketr(T ′k×nM +
1

2
(A′Tk×nB′)′A′Tk×nB′)

×
n
∏

i=1

k
∏

j=1

Φ(
α ji(A

′Tk×nB′)i j
Æ

(1+ (α2
ji
)
).

Hence the following definition and theorems.

Definition 4. (Matrix Variate Skew Normal Random Variable) Let

zi j ∼ 2φ(zi j)Φ(α jizi j)

for i = 1,2, . . . , n, and j = 1,2, . . . , k be independent univariate skew normal random variables.

Then the matrix variate random variable Z = (zi j) has density

2nkφk×n(Z)

n
∏

i=1

k
∏

j=1

Φ(α jie
′
j Zc i)



D. Akdemir, A.K. Gupta / Eur. J. Pure Appl. Math, 3 (2010), 128-140 136

where φk×n(Z) =
∏n

i=1

∏k

j=1φ(zi j), and e j and c i are the elementary vectors of the coordinate

system Rk and Rn respectively. Let A be a k×k constant matrix, B be a n×n constant matrix and

M be a k×n-dimensional constant matrix. A random variable X = AZB+M is distributed with re-

spect to matrix variate skew symmetric distribution with location parameter M , scale parameters

(A, B), and shape parameter ∆ = (α ji). We denote this by X ∼ MSNk×n(M ,A, B,∆). If the den-

sity exists it is given in Equation (11). We denote this case by writing X ∼ msnk×n(M ,A, B,∆).

Theorem 3. If X has multivariate skew-normal distribution MSNk×n(M ,A, B,∆) then the mo-

ment generating function of X evaluated at Tk×n is given by

MX (Tk×n) = 2nketr(T ′k×nM +
1

2
(A′Tk×nB′)′A′Tk×nB′)

×
n
∏

i=1

k
∏

j=1

Φ(
α ji(A

′Tk×nB′)i j
p

(1+α ji
2)
). (12)

By Definition 5 we can write Z ∼ MSNk×n(0, Ik, In,∆), and prove the following theorems.

Theorem 4. Assume that Y ∼ MSNk×n(M ,A, B,∆) and X = CY D+ N where C , D and N are

matrices of order k′×k, n×n′ and k′×n′ respectively. Then X ∼ MSNk′×n′(C M D+N , CA, BD,∆).

Proof. From assumption, we have Y = AZB+ M , and so X = CAZBD+ (C M D+ N), i.e.,

X ∼ MSNk′×n′(C M D+ N , CA, BD,∆).

Theorem 5. Let x 1, x 2, . . . x n be independent, where x i is distributed according to snk(0,Σ1/2,α).

Then,
n
∑

j=1

x ′ jΣ−1x j ∼ χ2
kn.

Proof. Let y ∼ Nk(µ = 0,Σ). Then y′Σ−1 y ∼ χ2
k
, and x ′ jΣ−1x j and y′Σ−1 y have the

same distribution from Theorem 2. Moreover, x ′ jΣ−1x j are independent. Then the desired

property is proven by the addition property of χ2 distribution.

It is well known that if X ∼ φk×n(M ,AA′,Ψ = In) then the matrix variate X X ′ has

the Wishart distribution with the moment generating function given as |(I − 2(AA′)T )|−n/2,

(AA′)−1− 2T being a positive definite matrix. The following theorem implies that the decom-

position for a Wishart matrix is not unique.

Theorem 6. If a k×n matrix variate random variable X has msnk×n(0k×n,A, In,∆) distribution

for constant positive definite matrix A of order k then X X ′ ∼Wk(n,AA′).

Proof. The moment generating function of the quadratic form X X ′ can be obtained as

follows, for any T ∈ Rk×k, with (AA′)−1 − 2T being a positive definite matrix,

E(etr(X X ′T )) =
∫

R
k×n

etr(X X ′T )dFX



D. Akdemir, A.K. Gupta / Eur. J. Pure Appl. Math, 3 (2010), 128-140 137

=

∫

R
k×n

2nketr(−1

2
(AA′)−1X X ′+ X X ′T )

∏n

i=1

∏k

j=1Φ(α jie
′
jA
−1X c i)

(2π)nk/2|A|n dX

=

∫

R
k×n

2nketr(−1

2
X ′((AA′)−1 − 2T )X )

∏n

i=1

∏k

j=1Φ(α jie
′
j
A−1X c i)

(2π)nk/2|A|n dX

=

∫

R
k×n

2nketr(−1

2
Z ′Z)
∏n

i=1

∏k

j=1Φ(α jie
′
jA
−1((AA′)−1− 2T )1/2Zc i)

(2π)nk/2|(I − 2(AA′)T )|n/2 dZ

=
2nk
∏n

i=1

∏k

j=1 Ez(Φ(cz))

(2π)nk/2|(I − 2(AA′)T )|n/2
(z ∼ φ(z) and c ∈ R is a constant)

=
2nk(1

2
)nk

|(I − 2(AA′)T )|n/2
= |(I − 2(AA′)T )|−n/2.

3. Generalized Matrix Variate Skew Normal Distribution

Definition 5. (Generalized Matrix Variate Skew Normal Density). Let X = AZB + M where A

(k× k), B (n× n) and M (k× n) are constant matrices and let A, B be nonsingular. We call the

density

2knφk×n(A
−1(X −M)B−1)

∏k

j=1 H(α jie
′
j
(A−1(X −M)B−1)c i)

|A|n|B|k (13)

the matrix variate skew normal density and denote this by gmsnH
k×n
(M ,A, B,∆).

Let Z ∼ gmsnH
k×n
(0k×n, Ik, In,∆). The moment generating function of Z evaluated at Tk×n

is MZ (Tk×n), it can be obtained as follows:

MZ (Tk×n) = EHk×n(Z)
(etr(T ′k×nZ)

n
∏

i=1

k
∏

j=1

H(α jie
′
j Zc i))

=
2k

(2π)k/2

n
∏

i=1

∫

Rk

e−
1

2
z′

i
z i+t i

′z i

k
∏

j=1

H(α jie
′
j z i)dz

=
2k

(2π)k/2

n
∏

i=1

∫

Rk

e−
1

2
(z′

i
z i−2t i

′z i)
k
∏

j=1

H(α jie
′
j z i)dz



D. Akdemir, A.K. Gupta / Eur. J. Pure Appl. Math, 3 (2010), 128-140 138

=

n
∏

i=1

2ke
1

2
t i
′ t i

(2π)k/2

∫

Rk

e−
1

2
(z′

i
z i−2t i

′z i+t i
′ t i)

k
∏

j=1

H(α jie
′
j z i)dz

=

n
∏

i=1

2ke
1

2
t i
′ t i

(2π)k/2

∫

Rk

e−
1

2
(z i−t i)

′(z i−t i)
k
∏

j=1

H(α jie
′
j z i)dz

=

n
∏

i=1

2ke
1

2
t i
′ t i

(2π)k/2

k
∏

j=1

∫

R

e−
1

2
(zi j−(t )i j)

2

H(α ji z i j)dzi j

=

n
∏

i=1

2ke
1

2
t i
′ t i

k
∏

j=1

∫

R

1
p

(2π)
e−

1

2
(yi j )

2

H(α ji y i j +α ji(t )i j)d yi j

=

n
∏

i=1

2ke
1

2
t i
′ t i

k
∏

j=1

H(
α ji(t )i j
p

(1+α ji
2)
)

= 2nketr(
1

2
Tk×n

′Tk×n)

n
∏

i=1

k
∏

j=1

H(
α ji(T )i j
p

(1+α ji
2)
).

Let X = AZB + M for constant (k× k) matrix A, (n× n) matrix B and k× n dimensional

constant matrix M . Then the moment generating function of X evaluated at Tk×n ∈ Rk×n is

MX (Tk×n), this can be obtained as follows:

MX (Tk×n) = 2nketr(T ′k×nM +
1

2
(A′Tk×nB′)′A′Tk×nB′)

×
n
∏

i=1

k
∏

j=1

H(
α ji(A

′Tk×nB′)i j
p

(1+α ji
2)
).

Hence the following definition and theorems.

Definition 6. (Matrix Variate Generalized Skew Normal distribution) Let

zi j ∼ 2φ(zi j)H(α jizi j)

for i = 1,2, . . . , n, and j = 1,2, . . . , k be independent univariate generalized skew normal random

variables. The matrix variate random variable Z = (zi j) has density

2nkφk×n(Z)

n
∏

i=1

k
∏

j=1

H(α jie
′
j Zc i)

where φk×n(Z) =
∏n

i=1

∏k

j=1φ(zi j), and e′j and c′i are the elementary vectors of the coordinate

system Rk and Rn respectively. Let A be a k × k constant matrix, B be a n× n constant matrix

and M be a k × n-dimensional constant matrix. Then the random matrix X = AZB + M is

distributed with respect to generalized matrix variate skew symmetric distribution with location



D. Akdemir, A.K. Gupta / Eur. J. Pure Appl. Math, 3 (2010), 128-140 139

parameter M, scale parameters (A, B), and shape parameter ∆ = (α ji). We denote this by

X ∼ GMSN H
k×n
(M ,A, B,∆). If the density exists it is given in Equation (13). We denote this case

by writing X ∼ gsnH
k×n
(M ,A, B,∆).

Theorem 7. If X has generalized matrix variate skew-normal distribution,

GMSN H
k×n
(M ,A, B,∆), then the moment generating function of X evaluated at Tk×n is given by

MX (Tk×n) = 2nketr(T ′k×nM +
1

2
(A′Tk×nB′)′A′Tk×nB′)

×
n
∏

i=1

k
∏

j=1

H(
α ji(A

′Tk×nB′)i j
p

(1+α ji
2)
). (14)

By Definition 6 we can write Z ∼ GMSN H
k×n
(0, Ik, In,∆), and prove the following theo-

rems.

Theorem 8. Assume that Y ∼ GMSN H
k×n
(M ,A, B,∆) and X = CY D + N where C , D and N

are matrices of order k′ × k, n × n′ and k′ × n′ respectively. Then X ∼ GMSN H
k′×n′(C M D +

N , CA, BD,∆).

Proof. From assumption we have Y = AZB + M , and so X = CAZBD + (C M D + N), i.e.,

X ∼ MSNk′×n′(C M D+ N , CA, BD,∆).

Theorem 9. Let x 1, x 2, . . . x n be independent, where x i is distributed according to gsnH
k
(0,Σ1/2,α).

Then,
n
∑

j=1

x ′ jΣ−1x j ∼ χ2
kn

.

Proof. Let y ∼ Nk(µ = 0,Σ). Then y′Σ−1 y ∼ χ2
k
, and x ′ jΣ−1x j and y′Σ−1y have the

same distribution from Theorem 2. Moreover, x ′ jΣ−1x j are independent. Then the desired

property is proved by the addition property of χ2 distribution.

4. Extensions

An odd function, say w(x), can be used to replace the term in the form α ji x in the skewing

function to give more flexible families of densities. We can take w(x ji) =
λ1 xp

1+λ2 x2
, to obtain

a matrix variate form of the skew symmetric family introduced by Arellanno-Valle at al. [1];

if we take w(x) = αx + β x3, we obtain a matrix variate form of the skew symmetric family

introduced by Ma and Genton [8]; or take w(x) = si gn(x)|x |α/2λ(2/α)1/2 to obtain a matrix

variate form of the skew symmetric family introduced by DiCiccio and Monti [5].

The skewing function of the matrix variate skew normal density in Equation (11), i.e.

k
∏

j=1

n
∏

i=1

Φ(α jie
′
j(A
−1(X −M)B−1)c i), (15)



REFERENCES 140

can be replaced by

Φk∗×n∗(ΓA−1(X −M)B−1Λ;0, C , D) (16)

where Γ, Λ, C (positive definite) and D (positive definite) are matrices of dimensions k∗ × k,

n × n∗, k∗ × k∗ and n∗ × n∗ correspondingly. In this case, the normalizing constant 2kn in

(11) will have to be changed to EZ (P(Y < ΓA−1(Z−M)B−1Λ|Z)) for Y ∼ φk∗×n∗(0, C , D) and

Z ∼ φk×n(M ,AA′, B′B). If all of Γ, Λ, C and D are positive definite diagonal matrices then

(16) can be written in the same form as (15), therefore the latter density is more general. Ar-

mando [2] introduced the matrix variate closed skew-normal distribution based on marginal

representation or hidden truncation. The density of the matrix variate closed skew-normal

distribution has the expression (16) as for its skewing function. The class of distributions

defined by msnk×n(M ,A, B,∆) is a subclass of the matrix variate closed skew-normal dis-

tribution. The extension gmsnH
k×n
(M ,A, B,∆) is not a subclass of the matrix variate closed

skew-normal distribution.

References

[1] R.B. Arellano-Valle, H.W. Gomez, and F.A. Quintana. A new class of skew-normal distri-

butions. Communications in Statistics-Theory and Methods, 33(7):1465–1480, 2004.

[2] J. Armando, G.F. Domínguez-Molina, R. Ramos-Quiroga, and A.K. Gupta. A matrix variate

closed skew-normal distribution with applications to stochastic frontier analysis. Commu-

nications in Statistics - Theory and Methods, 36(9):1691–1703, 2007.

[3] L. Bailey, T.F. Moore, and B.A. Bailar. An interviewer variance study for the eight im-

pact cities of the national crime survey cities sample. Journal of the American Statistical

Association, 73(1):16–23, 1978.

[4] J.T. Chen and A.K. Gupta. Matrix variate skew normal distributions. Statistics, 39(3):247–

253, 2005.

[5] T.J. DiCiccio and A.C. Monti. Inferential aspects of the skew exponential power distribu-

tion. Journal of American Statistical Association, 99(466):439–450, 2004.

[6] A.K. Gupta and D.K. Nagar. Matrix Variate Distributions. Chapman & Hall/CRC, 2000.

[7] S.W. Harrar and A.K. Gupta. On matrix variate skew-normal distributions. Statistics,

42(2):179–194, 2008.

[8] Y.Y. Ma and M.G. Genton. Flexible class of skew-symmetric distributions. Scandinavian

Journal of Statistics, 31(3):459–468, 2004.

[9] S. Zacks. Parametric Statistical Inference. Pergamon Press, New York, 1981.