A new construction for skew multivariate distributions

Transcription

1 Journal of Multivariate Analysis A new construction for skew multivariate distributions DipakK. Dey, Junfeng Liu Department of Statistics, University of Connecticut, 5 Glenbrook Road, U-40, Storrs, CT , USA Received December 00 Available online November 004 Abstract This paper considers a new approach to develop a very general class of skew multivariate distributions. The approach is based on a linear combination of an elliptically distributed random variable with a linear constraint. Using this approach two different classes of multivariate distributions are constructed based on original distribution. These new classes include different types of skew normal type A and type B and other skew elliptical distributions, exist in the literature. We also derive the moment generating function, marginal and conditional density of our proposed classes of distributions. Straightforward explanations are applied to demonstrate the relationships among previous approaches by others with our proposed class of skew distributions. 004 Elsevier Inc. All rights reserved. AMS 99 subject classification: 60E05; 6H0 Keywords: Elliptical distribution; Linear combination; Linear constraint; Moment generating function; Multivariate normal; Skewness. Introduction In many fields such as economics, psychology and sociology, sometimes error structures in a regression type models no longer satisfy symmetric property. Often there is a presence of high skewness. To preserve important properties it is natural to decompose some distributions into original symmetric portion and accumulated linearly constrained portion to Corresponding author. Fax: address: dey@stat.uconn.edu D.K. Dey X/$ - see front matter 004 Elsevier Inc. All rights reserved. doi:0.06/j.jmva

2 34 D.K. Dey, J. Liu / Journal of Multivariate Analysis demonstrate the prevalence of skewness. The first appearance of the skew normal distribution SN, hereafter dates backto Roberts [6], who considered Z minx, Y, X, Y is correlated bivariate normal random variable. After that several papers have been written on skew distributions, both on univariate and multivariate set up. They include Azzalini and Dalla Valle [5], who proposed Type A-MSN multivariate skew normal α, μ, Ω with density of the form fx; α, μ, Ω p x; μ, ΩΦα x μ, x R p. Recently Azzalini and Capitanio [4], Gupta et al. [] and Domínguez-Molina et al. [8] also studied multivariate skew normal distributions. Gupta et al. [] proposed Type B- MSN multivariate skew normal D, μ, Ω with density of the form fx; D,μ, Ω Φ p Dx μ;0,i Φ p D0;0,I+DΩD p x; μ, Ω, x Rp. The paper of Domínguez-Molina et al. [8] includes Gupta et al. [] as a special case. Recently Branco and Dey [7], and Sahu et al. [7] constructed two new classes of multivariate skew elliptical distributions. They used a conditional method by introducing positive random vector components. Whereas Branco and Dey [7] applied similar idea of Azzalini and Dalla Valle [5] to construct multivariate skew elliptical distribution. Liseo and Loperfido [4] constructed a broad class of multivariate skew normal distributions by location vector mixture within linear constraints. Gupta et al. [] and Sahu et al. [7] method is a special case of that of Liseo and Loperfido [4]. However, Domínguez-Molina et al. [8] is not a special case of that of Liseo and Loperfido [4]. Jones [3] applied the marginal replacement method in constructing skew multivariate distributions with application to skewing spherically symmetric distributions. Jones method is a quite self-contained and flexible way to construct skew distributions. To preserve some properties of the original symmetric distributions, Jones concentrated on two-dimensional skew construction and proposed only one component replacement for multivariate skew distributions. Skewness construction by truncation was discussed in Arnold et al [], subsequently Arnold and Beaver [] gave a comprehensive review over the literatures on multivariate skewness construction, interpretation and characterizing property. Their explanations for skewness mechanism is related to hidden truncation and/or selective reporting. Considering a p-dimensional random vector W which has an elliptically contoured distribution, e.g., W μ + Σ / Z, Z is a spherically symmetric random vector, μ is a location vector and Σ is a scale matrix. Such a random vector Z may be represented as Z RU,...,U p, R and u are independent random variables, with PR > 0 and R F R, R is uniformly distributed over the unit p-sphere. The distribution of Z determines the distribution of W. Consequently the components of W will be dependent, so are the components of Z. It is known that the components of Z will be independent only if R has a particular chidistribution, i.e., R is a constant multiple of the square root of a χ distribution with p degrees of freedom. Then Z will be N0,Iand W will also have independent components if Σ / is orthogonal. Starting with a p+-dimensional elliptically contoured random vector of the form, say W 0,...,W p, Arnold and Beaver [] considered conditional distribution of W,...,W p given that W 0 >c, which they called a p-dimensional skew-elliptical density. If we consider a special case in which Z has an appropriate chi-distribution, this

3 D.K. Dey, J. Liu / Journal of Multivariate Analysis skewed model will reduce to the model obtained by applying transformations to the density [ p ] fw; 0, λ ψw i Ψλ w, i ψ is the pdf, Ψ is the cdf. They chose Z such that U has Cauchy marginal to get skewed- Cauchy distribution. Suppose W,...,W p and U are independent random variables with densities given by ψ w,...,ψ p w p and cdf Ψ 0 u. The conditional distribution of W given that λ 0 + λ W>U, λ 0 Rand λ R p is p i f W A w ψw iψ 0 λ 0 + λ w, P A A {λ 0 + λ W>U}. If we assume a joint density of W as ψw, the above density will be f W A w ψwψ 0λ 0 + λ w. P A Arnold and Beaver [] point out that, if we begin with ψw an elliptically contoured density then this preceding formula included the Branco and Dey [7] approaches as special cases. A different type of skew-elliptical distribution by Azzalini and Capitanio [4] is also a special case of it. In this paper, we propose a new approach based on linear constraints and linear combination LCLC to develop a new class of skew-elliptical distributions. This class includes Liseo and Loperfido [4] and Domínguez-Molina et al. [8] density in a more general sense. We also derive the corresponding density function for each case of LCLC skewness constructions. Liseo and Loperfido [4] claim Type B-MSN density does not generalize Type A-MSN density. We illustrate the relationship between Type A-MSN and Type B-MSN and point out that all of the aforementioned skewness constructions are embedded within our scheme although ours is a dichotomous construction. The format of the paper is as follows. Section develops our main results on a more general constructions for multivariate skew distribution. Section 3 summarizes all of the special cases of our construction. They include previous authors results on multivariate skew density functions to date. Section 4 develops the moment generating function, marginal density and conditional density and some properties of our multivariate skew distribution, especially multivariate skew normal distribution. Section 5 provides some concluding remarks.. Main results.. Elliptical distributions Consider a p-dimensional random vector X having probability density function pdf of the form fx μ, Ω; g p Ω g p x μ Ω x μ, x R p,

4 36 D.K. Dey, J. Liu / Journal of Multivariate Analysis g p u is a non-increasing function from R + to R + defined by g p u Γp/ gu; p π p/ rp/ gr; p dr 0 and gu; p is a non-increasing function from R + to R + such that the integral 0 rp/ gr; p dr exists. In this paper, we will always assume the existence of the pdf fx μ, Ω; g p. The function g p is often called the density generator of the random vector X. Note that the function gu; p provides the kernel of X and other terms in g p constitute the normalizing constant for the density f. In addition the function g, hence g p, may depend on other parameters which would be clear from the context. For example, in case of t distributions the additional parameter will be the degrees of freedom. The density f defined above represents a broad class of distributions called the elliptically symmetric distribution and we will use the notation X Elθ, Ω; g p, henceforth in this article. Let Fx θ, Ω; g p denote the cumulative density function cdf of X X Elθ, Ω; g p. We consider two examples, namely the multivariate normal and t distributions. Example Multivariate normal. Let gu; p exp u/. Then straightforward calculation yields g p u e u/ π p/. Then fx μ, Ω; g p π p/ Ω exp x μ Ω x μ,x R p, which is the pdf of the p-variate normal distribution with mean vector θ and covariance matrix Ω. We denote this distribution by N p θ, Ω and the pdf by N p x θ, Ω henceforth. Example Multivariate t. Let gu; p, ν + u ν+p/, ν > 0. ν Here g depends on the additional parameter ν, the degrees of freedom. Then straightforward calculation yields ν+p Γ g p u; ν Γ ν gu; p, ν. νπp/ Hence fx μ, Ω; g p Γ ν+p Γ ν νπp/ Ω

5 D.K. Dey, J. Liu / Journal of Multivariate Analysis x μ Ω ν+p/ x μ,x R p, ν which is the density of the p-variate t distribution with parameters θ, Ω and degrees of freedom ν. We denote this distribution by t p,ν θ, Ω and the density by t p,ν x θ, Ω henceforth. The subscript p will be omitted when it is equal to. The following lemma will be useful for the rest of the paper. Lemma.. If X Elμ, Ω; g p, and X is partitioned as X X,X, X is p and X is p with X μ Ω Ω El, Ω ; g X μ Ω Ω p,p p + p then X X x Elμ., Ω. ; g p qx, μ. μ + Ω Ω x μ, Ω. Ω Ω Ω Ω, qx x θ Ω x θ, g p a u Γp / ga+u;p π p/ 0 rp / ga+r;p dr, g a u gp+ u+a. g p a Proof. The proof follows from Fang et al. []. Lemma.. Suppose X Elμ, Ω,g p, and R is a matrix of order k p, then RX ElRμ,RΩR,g k... General skew multivariate elliptical distribution Suppose the random vector Y Elθ, Ω; g p and satisfies the following linear constraint: RY + d 0, R is a given matrix of dimension k p, k p and d is a vector of dimension k. Then defining p c as the probability of the constraint set, we have p c PRY + d 0 PRY Rμ d Rμ F d Rμ 0,RΩR,g k, F is the cdf of an elliptical distribution with location 0 k, scale RΩR, density generator g k, and 0 k is a k-dimensional vector of 0.

6 38 D.K. Dey, J. Liu / Journal of Multivariate Analysis Further partitioning R as R R,R, we can express constraint as R Y + R Y + d 0, 3 the dimension of the vectors of Y and Y are, respectively, p and p. The dimension of R is k p, the dimension of R is k p. Here R is of full row rank, and R is an arbitrary matrix such that the multiplication matrix is valid.... Linear constraint and linear combination of type- LCLC Suppose the dimension of Y is not equal to the dimension of Y, i.e., p p. Consider the distribution of C Y, C is a non-singular square matrix with dimension p p. Here Y is used only as an auxiliary variable for producing skewness for Y, it does not show up in the final form of the skew distribution. Theorem... Under the linear conditions in LCLC, given a non-singular p p matrix C, the density function for X C Y under constraint 3 is F d R μ Ω Ω μ R + R Ω Ω C x 0, Ω x,g x k F d Rμ 0,RΩR,g k fx C μ,c Ω C,gp, 4 k is the number of rows in R, { Ωx R Ω Ω Ω Ω R g k x g k C x μ Ω C x μ. Proof. Under linear constraint 3, the joint density function of Y,Y is fy,y μ, Ω; g p f c y,y F d Rμ 0,RΩR ; g k on R Y + R Y + d 0. Integrating out Y, we get the density of Y as f c y F d Rμ 0,RΩR ; g k F d Rμ 0,RΩR ; g k fy μ, Ω ; g p dy R Y d R Y fy,y μ, Ω; g p dy R Y d R Y fy,y μ, Ω; g p fy μ, Ω ; g p F d Rμ 0,RΩR ; g k fy μ, Ω ; g p fy y, μ, Ω; g p dy. R Y d R Y Now using Lemma., Y Y Elμ., Ω. ; g p qy,

7 D.K. Dey, J. Liu / Journal of Multivariate Analysis p is the dimension of Y. Now using Lemma., we have R Y Y ElR μ.,r Ω. R ; gk qy. Thus f c y F d Rμ 0,RΩR ; g k fy μ, Ω ; g p fy y, μ, Ω; g p dy R Y d R Y F d Rμ 0,RΩR ; g k fy μ, Ω ; g p F d R y R μ + Ω Ω y μ, R Ω Ω Ω Ω R, g k y μ Ω y μ. The final result follows after some simplifications.... Linear constraint and linear combination of type- LCLC If the dimension of Y is same as that of Y, i.e., p p, we consider the distribution of C Y + C Y under the constraint 3, further R C R C is full row rank, C,C are non-singular square matrices. Theorem... Under the conditions in LCLC, the pdf for X C Y + C Y is F d C 3 μ wx Ω wx Ωwx μ wx R C + C 3 Ω wx F d Rμ 0,RΩR ; g k Ωwx x 0,C 3 Ω C 3 ; gk a fx μ wx, Ωwx ; gp, 5 μ wx μ wx μ wx Ω wx Ω wx Ωwx Ω wx Ωwx C μ C μ + C μ C 3 R C R C, μ μwx + Ω wx Ωwx x μ wx, Ω Ωwx Ωwx Ωwx Ω wx, a x μ wx Ω wx x μ wx., C 0 Ω Ω C C C C Ω Ω 0 C, C Ω C C Ω C + C Ω C C Ω C + C Ω C C Ω C + C Ω C + C Ω C + C Ω C Proof. Consider the transformation { { W C Y, X C Y + C Y, Y C W, Y C X W.,

8 330 D.K. Dey, J. Liu / Journal of Multivariate Analysis Clearly this is a one-to-one mapping. Now it follows that f X x fx,wdw p c fy x, w, y x, w y,y p c R Y +R Y d x, w dw fx,wdw p c R C W+R C X W d fx,wdw p c R C R C W d R C X fxfw xdw p c R C R C W d R C X fx fw xdw. p c C 3 W d R C X Further we know W C Y μ wx El X C Y + C Y μ wx, Ω wx ; g p. Now recall the properties of multivariate elliptical distribution. From Lemmas. and., it follows that W X Elμ, Ω ; gp a, C 3 W X ElC 3 μ,c 3Ω C 3 ; gk a, X Elμ wx, Ωwx ; gp. Thus we have f X x p c fx μ wx, Ωwx ; gp F d R C x C 3μ,C 3Ω C 3 ; gk a a F d R C x C 3μ,C 3Ω C 3 ; gk F d Rμ 0,RΩR ; g k The final result follows after some simplifications. fx μ wx, Ωwx ; gp. Remark.. The following notations are introduced to define the corresponding dimensions: X p, μ p, Ω p p, R k p, d k, C p p, g p, C p p, C p p. These notations will be used throughout the rest of the paper. For simplicity we will not identify dimensions of all the variables. i From the proof of Theorem.., we can see that if R C R C has full row rank, then LCLC is a reparameterization of LCLC. Although from the construction point of view, they are of different classes, they enjoy same distribution properties under mild conditions.

9 D.K. Dey, J. Liu / Journal of Multivariate Analysis ii We will use notation X GSMEμ, Ω,R,d,C,g p Theorem.. to represent Type -GSME Generalized Skew Multivariate Elliptical distribution. iii We will use notation X GSMEμ, Ω,R,d,C,C,g p Theorem.. to represent Type -GSME Generalized Skew Multivariate Elliptical distribution. In this case the vector dimension combination is p,p, with p p. iv In view of the reparameterization equivalence property of LCLC and LCLC, it is always good practice to focus on the evolved distribution of Y or Y + Y under an appropriately chosen original distribution plus linear constraint, e.g., the full-ranksquare matrix C in i, C and C in iii are redundant during the skew construction procedure. To avoid overparametrization and non-identifiability in estimation procedures we suggest imposing C I, and C C I. Example 3. Let X p X p,x p be LCLC or LCLC multivariate skew normal distribution with density Φ p b μ Δ, Ω Δ Φ ka Bx μ, Ω p x μ, Ω then the marginal density function for X p is f X x Φ p b μ Δ, Ω Δ p x μ, Ω Φ k a + B Ω Ω μ μ μ B + B Ω Ω x ; 0,B Ω Ω Ω Ω B + Ω, B B k,p,b k,p, μ p p p + p,k k + k, Ωp,p Ω Ω p,p Ω p,p Ω p,p μp μ p μ, μ k k μ, k, Ω Ω k,k Ω k,k Ω k,k Ω k,k Proof. It follows that f X x fx,x dx gx gx x dx p x ; μ, Ω Φ p b; μ Δ, Ω Δ Φ p a Bx; μ, Ω p x ; μ + Ω Ω x μ, Ω Ω Ω Ω dx {Recall p x,x ; μ, Ω p x ; μ, Ω p x ; μ + Ω Ω x μ, Ω Ω Ω Ω } p x ; μ, Ω Φ p b; μ Δ, Ω Δ Φ p a B x B x ; μ, Ω p x ; μ + Ω Ω x μ, Ω Ω Ω Ω dx.

10 33 D.K. Dey, J. Liu / Journal of Multivariate Analysis {Recall B B,B and x x,x } p x ; μ, Ω Φ p b; μ Δ, Ω Δ Φ k B + B Ω Ω x ; μ a +B μ Ω Ω μ, B Ω Ω Ω Ω B + Ω. Remark.. The formula is same for both LCLC and LCLC setups, although they are developed from different skew mechanisms..3. Closure of marginal and conditional distribution in LCLC In this section, we study the closure properties of the marginal and conditional distributions under the linear constraint and linear combination of first type. Theorem.3.. The marginal density is closed under LCLC setup. Proof. Suppose Z ν El Y Y μ μ, Ω,g p. We express linear constraints as R Z + R Y + d 0, Y Y,Y and consider random vector C Y. Let X C, X C X Y Y, C, then Z X ν I 0 I 0 El, Ω C μ 0 C 0 C,g p. Recall the relationship X C X Y, Y C X,R X Y R C X. X Based on the dimensions of X and X, let R 3 R 3, R 3, R C, then the linear constraint R Z + R Y + d 0 is equivalent to Z R R 3, + R 3, X + d 0. X

11 D.K. Dey, J. Liu / Journal of Multivariate Analysis Thus the marginal distribution of X is SMEμ, Ω,R,d,C,g p, the parameters are R R,R 3,, R R 3,, R R,R, d d, μ ν,c, μ,c, μ, I 0 I 0 Ω Ω 0 C 0 C, C I, g g. Theorem.3.. The conditional density is closed under LCLC setup. Proof. Suppose Z ν Z Y Y El Y μ μ Ω Ω, Ω Ω Ω,g p We express linear constraint as R Z + R Y + d 0 and consider random vector C Y. Let Σ Σ Ω Ω yy C, y,c Σ Σ y, C, y the dimension of C, is p p, the dimension of C, is p p, such that p p + p. Then Z Z X C, Y X C, Y ν Ω Ω C, Ω C, El C, μ, C, Ω C, Ω C, C,Ω C,,g p. C, μ C, Ω C, Ω C, C,Ω C, Using the property of elliptical distribution we get Z X x Elμ, Ω,g p +p, X ν μ Ω C +, C, μ C, Ω C, Ω Ω Ω C, C, Ω C, Ω C, Ω C, C, Ω C,. C, Ω C, x C, μ, C, Ω C, C, Ω C, Ω C,.

12 334 D.K. Dey, J. Liu / Journal of Multivariate Analysis Using same partition as in Theorem.3., suppose R 3 R 3, R 3, R C. Then the linear constraint R Z + R Y + d 0 is equivalent to Z R R 3, + R 3, X + d 0. X Thus the conditional distribution of X given X is SMEμ, Ω,R,d,C,C,g p +p, R R, R R 3,, R R,R, d R 3, a + d, C I, μ μ, Ω Ω, g g. 3. Special cases In this section we consider several examples from the literature and show how they can be constructed as a special case from LCLC or LCLC approaches. 3.. Type A-MSN α, μ, Ω Azzalini and Dalla Valle [5] constructed multivariate skew normal density of the form fx; α, μ, Ω p x; μ, ΩΦα x μ, x R p. Their method is included in our LCLC setup by choosing R, R 0 p, R R,R, d 0, C I p, μ 0, μ, δ... δ p, δ Ω. Ω, δ k gμ exp μ/. 3.. Type B-MSN D, μ, Ω Gupta et al. [] constructed multivariate skew normal density of the form fx; D,μ, Ω Φ pdx μ;0,i Φ p D0;0,I+DΩD p x; μ, Ω, x Rp. Their method is included in our LCLC setup

13 D.K. Dey, J. Liu / Journal of Multivariate Analysis by choosing R I q q, R D q p, R I q q, D q p, d 0 q, C I p p, μ Dμ, μ, Iq q 0 Ω, 0 Σ p p gμ exp μ/ Liseo and Loperfido class of MSN Liseo and Loperfido [4] constructed a broader class of multivariate skew normal distribution which includes both Type A-MSN as well as Type B-MSN as special cases. The density of their skew normal distribution is Φ k 0; CΔΣ x + Ω μ + d,cδc Φ k 0; Cμ + d,cωc p x, μ, Σ + Ω, Δ ΣΣ + Ω Ω. It is a special case of LCLC with μ Ω 0 μ, Ω,R 0 0 Σ C, R 0,C I p p,c I p p. Another way to illustrate Liseo and Loperfido [4] method is to consider X X 0 NX 0, Σ and X 0 Nμ, Ω with linear constraint K X 0 + d 0, the dimension of K is p m, m p, such that K is full rank. The random variable X can be regarded as the sum of two independent random variables, of which one is a N p μ, Ω distribution with linear constraint K X + d 0, the other is a N p 0, Σ distribution. Liseo and Loperfido [4] claim that when k, their approach is a slight generalization of the Type A-MSN distribution, as when k p, their approach produces a representation of the Type B-MSN density. Here we observe that the Type A-MSN and Type B-MSN are both derived from LCLC, as Liseo and Loperfido [4] construction is derived from LCLC Gupta, González-Farías and Domínguez-Molina class of MSN Domínguez-Molina et al. [8] constructed a more general class Generalized Multivariate Skew Normal, or GMSN which includes Type B-MSN as a special case. The density is of the form f p,q x; μ, Σ,D,ν, Δ Φ q Dx; ν, Δ Φ q Dμ; ν, Δ + DΣD p x; μ, Σ. 6

14 336 D.K. Dey, J. Liu / Journal of Multivariate Analysis They used notation X SN p,q μ, Σ,D,ν, Δ. Ifwetakeν Dμ and Δ I p then the density reduces to f p,q x; μ, Σ,D,Dμ,I p Φ p Dx μ; 0,Ip Φ p 0; 0,I + DΣD p x; μ, Σ Type B-MSND, μ, Ω. Note that, Domínguez-Molina et al. [8] is a reparameterization of LCLC. To obtain their results, we only take normal distribution in our elliptical distribution framework and define Z ν Δ 0 MN, X μ 0 Σ Z I D 0 X Δ 0 and calculate the marginal density of X, i.e., R I q q, R D q p, Ω. The 0 Σ calculation of the density of X given Z DX 0 produces Domínguez-Molina et al. [8] results. On the other hand, for normal case, density 4 can be written as GMSN with parameters μ C μ, Σ C Ω C, D R + R Ω Ω C, ν R μ + d R Ω Ω μ, Δ Ω x R Ω Ω Ω Ω R. It is easy to checkthat Δ + DΣD RΩR. Similarly, density 5 can be written as GMSN with parameters μ μ wx, Σ Ωwx, D R C + C 3 Ω wx Ωwx, ν C 3 μ wx Ω wx Ωwx μ wx + d, Δ C 3Ω C 3, C 3 R C R C, μ wx C μ, C μ + C μ Ω wx Ω wx Ωwx C 0 Ω Ω Ω wx Ωwx C C C C Ω Ω 0 C, C Ω C C Ω C + Ω C C Ω + C Ω C C Ω + C Ω C + C Ω + C Ω C, Ω a Ωwx Ωwx Ωwx Ω wx, and x μ wx Ω wx x μ wx Skew elliptical distribution Here we consider two types of skew elliptical distributions and demonstrate how they can be obtained from our general formulation Skew elliptical distribution SE k μ, Ω, δ; g k+ Branco and Dey [7] consider Y Y 0,Y,Y,...,Y k El k+ μ, Σ;, μ 0, μ, μ μ,...,μ k, is the characteristic function with the scale parameter matrix Σ having the form δ Σ δω

15 D.K. Dey, J. Liu / Journal of Multivariate Analysis with δ δ,...,δ k. Let X [Y Y 0 > 0], Y Y,Y,...,Y k, then f X x; μ, Ω, δ; g k+ f g kf gqx λ x μ, x R p, f g k is the pdf of an elliptical distribution with generator function g k and F gqx is the cdf of a univariate elliptical distribution with g qx as the generator function. It includes Type A-MSN α, μ, Ω as a special case. It is a special case of LCLC with R, R 0, 0,...,0, Y Y 0, Y Y,Y,...,Y k, d 0 and C I k k Skew elliptical distribution SEμ, Σ,D; g m Sahu et al. [7] use the following construction. Suppose ε and Z are two m-dimensional random vectors. Let μ be an m-dimensional vector and Σ be an m m positive definite matrix. Assume that ε Y El Z μ 0, Σ 0 ; g m, 0 I 0 is the null matrix and I is the identity matrix. They consider a skew elliptical class of distributions by using the transformation X DZ + ε, D is a diagonal matrix with elements δ,...,δ m. Let δ δ,...,δ m. The class is developed by considering the random variable [X Z >0] Z>0 means Z i > 0 for i,...,m. Then f X x; μ, Σ,D; g m m f X x μ, Σ + D ; g m F I DΣ+D D DΣ + D x μ 0,I; g m qx μ, g a m u Γm/ ga + u; m π m/,a >0 0 rm/ ga + r; m dr and qx μ x μ Σ + D x μ. This density matches with the one obtained by Branco and Dey [7] only in the univariate case. The derived skew normal distribution is fx μ, Σ,D m Σ + D m Σ + D x μ Φ m I DΣ + D D DΣ + D x μ, m and Φ m denote, respectively, the density and cdf of an m-dimensional normal distribution with mean 0 and covariance matrix identity. Clearly this is different from

16 338 D.K. Dey, J. Liu / Journal of Multivariate Analysis Type A-MSN distribution. Note that this is a special case of LCLC with 0m Im m 0 μ, Ω,R μ m 0 Σ I m m,r 0,d 0 m, m m C D m m and C I m m. Remark 3.. We give a straightforward demonstration by LCLC parameterization for the closure of the marginal distribution in Domínguez-Molina et al. [8] class. In 6, X SN p,q μ, Σ,D,ν, Δ is partitioned into two components, X and X,of dimensions k and p k space, respectively. Then the marginal distribution of X is SN k,q μ, Σ,D + D Σ Σ, ν + D Σ Σ μ μ, Δ +D Σ Σ Σ Σ D, and μk νk Σk k Σ μ, ν, Σ k p k μ p k ν p k Σ p k k Σ p k p k D D q k D q p k. Domínguez-Molina et al. [8] use moment generating function and lengthy algebraic derivation Appendix A to get the marginal density. Here we use our LCLC set up to obtain the result in a straightforward way. Let X Z, X A,A, Y Z A Σ Σ. Y Y 3 A A The constraint Z DA 0 is equivalent to Y D Y D +D Σ Σ Y 3 0. Further it is easy to see that Y D Y,D + D Σ Σ Y 3 are independent. Now VarY D Y Δ + D Σ. D, VarY 3 Σ, D D + D Σ Σ, ν ν + D Σ Σ μ μ, and μ μ. Thus the marginal density for A is SN k,q μ, Σ,D + D Σ Σ, ν + D Σ Σ μ μ, Δ + D Σ. D. Remark 3.. We give a straightforward proof by LCLC parameterization for the closure of conditional distribution in Domínguez-Molina et al. [8] class.

17 D.K. Dey, J. Liu / Journal of Multivariate Analysis Suppose X SN p,q μ, Σ,D,ν, Δ and X is partitioned in two components, X and X, of dimensions k and p k, respectively. Then the conditional distribution of X given X is SN k,q μ + Σ Σ x μ, Σ Σ Σ Σ,D, ν D x, Δ, μk νk Σk k Σ μ, ν, Σ k p k μ p k ν p k Σ p k k Σ p k p k and D D q k D q p k. Domínguez-Molina et al. [8] use lengthy algebraic derivations to get the conditional density. Clearly using our LCLC set up we obtain the result in a straightforward way. Let the original random vector be partitioned as Z,X,X, and Z ν X N μ Δ 0, 0 Σ X μ then Z X X x N ν μ + Σ Σ x μ Δ 0,. 0 Σ. Conditionally, Z and X given X are still independent. Under the constraint Z DA X Z D,D 0 if and only if Z D X X D X 0. Now given X Δ VarZ D X X x, Σ VarX X x Σ Σ Σ Σ, D D, ν ν D X, and μ μ + Σ Σ x μ. Thus we get the conditional density for X X x under the linear constraint. 4. Moment generating functions First, we develop moment generating function of both LCLC and LCLC class of skew normal distribution. Lemma 4.. Let U be a k-dimensional vector and let B be a k p matrix. If V N p μ, Σ, then [ E V Φk u + BV; μ, Ω ] Φ k u μ + Bμ ; 0,BΣB + Ω. Proof. The proof is given in Box and Tiao [6].

18 340 D.K. Dey, J. Liu / Journal of Multivariate Analysis Lemma 4.. x μ Σ x μ t x x μ Σt Σ x μ Σt μ t t Σt. Proof. The proof follows from direct calculation. Theorem 4.. Under LCLC condition, the moment generating function of X is m x t Φ d Rμ 0,RΩR expμ C t + t C Ω C t Φ p d R μ Ω Ω μ R R Ω Ω μ + Ω C t; 0,R R Ω Ω Ω R R Ω Ω + R Ω Ω Ω Ω R. Proof. The proof follows from direct calculation using the pdf of X having form 3. Theorem 4.. Under LCLC condition, the moment generating function of X is m x t Φ d Rμ 0,RΩR expμwx t + t Ω wx t Φ p d C 3 μ wx Ω wx Ωwx μ wx R C C 3 Ω wx Ωwx μ wx + Ω wx t; 0,R C C 3 Ω wx Ω wx R C C 3 Ω wx Ωwx +C 3 Ω wx Ωwx Ωwx Ω wx C 3. Ωwx Proof. The proof follows from direct calculation using the pdf of X having form 4. Remark 4.. In LCLC setup, let X a GSNμ a, Ω a,r a,d a,c xa, X b GSNμ b, Ω b, R b,d b,c xb. Further suppose X a is independent of X b. Then the moment generating function of the joint distribution of X a,x b is obtained by computing M Xa,X b t M Xa,X b t a,t b Ee t a X a+t b X b M Xa t a M Xb t b Φ d Rμ 0,RΩR expμ C x t + t C x Ω C x t Φ p d R μ Ω Ω μ R R Ω Ω μ + Ω C x t; 0,R R Ω Ω Ω R R Ω Ω +R Ω Ω Ω Ω R, μa Ωa 0 Ra 0 da μ, Ω,R,d, μ b 0 Ω b 0 R b d b

19 C x D.K. Dey, J. Liu / Journal of Multivariate Analysis Cxa 0 ta,t. 0 C xb t b Hence the joint distribution of X,X is GSNμ, Ω,R,d,C x. Remark 4.. In LCLC setup, let X a GSNμ a, Ω a,r a,d a,c a,c a, and X b GSN μ b, Ω b,r b,d b,c b,c b. Further suppose X a is independent of X b. Then the moment generating function of the joint distribution of X a,x b is obtained by computing M Xa,X b t M Xa,X b t a,t b Ee t a X a+t b X b M Xa t a M Xb t b Φ d Rμ 0,RΩR expμwx t + t Ω wx t Φ p d C 3 μ wx Ω wx Ωwx μ wx R C C 3 Ω wx Ωwx μ wx + Ω wx t; 0,R C C 3 Ω wx Ωwx Ω wx R C C 3 Ω wx Ωwx μ μa μ b, Ω C Ca 0,C 0 C b +C 3 Ω wx Ωwx Ωwx Ω wx C 3, Ωa 0 Ra 0 da,r,d, 0 Ω b 0 R b d b Ca 0 ta,t, 0 C b t b hence the joint distribution of X,X is GSNμ, Ω,R,d,C,C. Remark 4.3. González-Farías et al. [9,0] formalized closed skew normal CSN family 6 and showed it is closed under full row ranklinear transformations and full column ranklinear transformations defining the singular skew normal distribution. The closure property also applies for LCLC/LCLC construction under reparameterization Section 3.4. Remark 4.4. In LCLC setup, suppose X GSNμ, Ω,R,d,C x, and A is a non-singular square matrix, then AX GSNμ, Ω,R,d,AC x, which means all the parameters are the same except C is replaced by AC. Thus any permutation of X X,X,...,X p is closed within LCLC setup. Remark 4.5. In LCLC setup, suppose X GSNμ, Ω,R,d,C,C, and A is a nonsingular square matrix, then AX GSNμ, Ω,R,d,AC,AC

20 34 D.K. Dey, J. Liu / Journal of Multivariate Analysis which means all the parameters are the same except C and C are replaced by AC and AC, respectively. Remark 4.6. In LCLC setup, suppose X GSNμ, Ω,R,d,C x, and A is a non-singular square matrix, then AX + b GSNμ, Ω,R,d,C x, μ μ μ μ, AC x μ + b Ω Ω Ω Ω Ω Ω C A Ω ACΩ ACΩ C A, R R R R R Cx, A d d R Cx A b, and C x I. It is clear to see that the representation is not unique based on the five parameters. Remark 4.7. In LCLC setup, if X GSNμ, Ω,R,d,C,C, and A is a non-singular square matrix, then AX + b GSNμ, Ω,R,d,C,C, μ μ μ + pac μ b μ + qac b Ω Ω Ω Ω Ω, Ω R R R, R, d d pr AC b qr AC b, C AC,C AC. The representation is again not unique based on the five parameters at least for any p + q. Now the following two theorems provide distribution of quadratic form under LCLC and LCLC setup through the moment generating functions. Theorem 4.3. In LCLC setup, suppose X GSNμ, Ω,R,d,C x and A is an symmetric idempotent matrix, then the mgf of X AX is given as M X AXt Φ p d Rμ; 0,RΩR I tac Ω C / and

21 D.K. Dey, J. Liu / Journal of Multivariate Analysis Proof. Recall the identity in Searle [8], exp μ Ω C I I tac Ω C C μ Φ p d R μ Ω Ω μ R + R Ω Ω C μ C I tac Ω C ; 0,R + R Ω Ω Ω C I tac Ω C C R + R Ω Ω + Ω x. exptx Ax p x; μ, Σ I taσ / exp μ Σ I I taσ p x; μ I taσ, ΣI taσ and recall the LCLC density function in Theorem.. and identity in Lemma 4.. The result follows after some simplifications. Theorem 4.4. In LCLC setup, suppose X GSNμ, Ω,R,d,C,C, and A is an symmetric idempotent matrix, then the mgf of X AX is M X AXt Φ p d Rμ; 0,RΩR I taω wx / exp Φ p d C 3 μ wx Ω wx R C + C 3 Ω wx +C 3 Ω wx Ωwx +C 3 Ω wx Ωwx μwx Ωwx μ wx Ω wx I I taω wx Ωwx I taω wx μ wx ; 0,R C Ω wx I taωwx R C + C 3 Ω C 3. Proof. The proof follows using similar idea used in the previous proof. For scale mixture of normal class of general elliptical distributions such as multivariate t-distribution, the moment generating function of both X and quadratic forms can be represented as an integral with respect to a mixing distribution. Further details are given in Liu and Dey [5]. 5. Concluding remarks The new class of skewed distributions obtained in this article is very general, quite flexible and widely applicable. Even for minx, Y case in Roberts [6], X, Y was correlated bivariate normal random variable, it can easily be seen to be a mixture of two univariate skew normal densities on two regions X minx, Y on X Y and Y minx, Y on Y X. Thus our construction based on linear combination with linear constraint is very general one and includes different types of skew distributions exist in the

22 344 D.K. Dey, J. Liu / Journal of Multivariate Analysis literature. Although the associated density functions are quite difficult to handle we can show that many special cases such models can be easily fit using MCMC methods. This paper creates a theoretical foundation on general skew elliptical distribution and is anticipated that further theoretical development relating to the distribution of ratio of quadratic forms, robustness study of Student s t-test and F-test can be obtained under this general skew elliptical distributions. Acknowledgments The authors thankthe editor and two referees for helpful comments which improved an earlier version of the work, especially the reparameterization demonstration in Section 3.4. and some corrections in Remark4.3. References [] B.C. Arnold, R.J. Beaver, R.A. Groeneveld, W.Q. Meeker, The nontruncated marginal of a truncated bivariate normal distribution, Psychometrika [] B.C. Arnold, R.J. Beaver, Skew multivariate models related to hidden truncation and/or selective reporting, Test Sociedad de Estadística e Investigación Operativa. [3] A. Azzalini, A class of distributions which includes the normal ones, Scand. J. Statist [4] A. Azzalini, A. Capitanio, Statistical applications of the multivariate skew normal distribution, J. Roy. Statist. Soc. B [5] A. Azzalini, A. Dalla Valle, The multivariate skew-normal distribution, Biometrika [6] G. Box, G. Tiao, Bayesian Inference in Statistical Analysis, Wiley, New York, 99. [7] M. Branco, D. Dey, A general class of multivariate skew-elliptical distributions, J. Multivariate Anal [8] J. Domínguez-Molina, G. González-Farías, A. Gupta, A general multivariate skew normal distribution, Technical Report No.0-09, Department of Mathematics and Statistics, Bowling Green State University, 00. [9] G. González-Farías, J. Domínguez-Molina, A. Gupta, Additive properties of skew normal random vectors, J. Statist. Plann. Inference, [0] G. González-Farías, J. Domínguez-Molina, A. Gupta, The multivariate closed skew normal distribution, Technical Report No.03-, Department of Mathematics and Statistics, Bowling Green State University, 003. [] A. Gupta, G. González-Farías, J. Domínguez-Molina, A multivariate skew normal distribution, Working paper I-0-9, Centro de Investigación en Matemáticas, A.C. Guanajuato, México, 00. [] K. Fang, S. Kotz, K. Ng, Symmetric Multivariate and Related Distributions, Chapman & Hall, London, 990. [3] M.C. Jones, A skew t-distribution, in: C.A. Charalambides, M.V. Koutras, N. Balakrishnan Eds., Probability and Statistical Models with Applications, Chapman & Hall/CRC, London, 00, pp [4] B. Liseo, N. Loperfido, A Bayesian interpretation of the multivariate skew-normal distribution, Statist. Probab. Lett [5] J. Liu, D.K. Dey, Skew-elliptical distributions, in: M.G. Genton Ed., Skew-Elliptical Distributions and their Applications: A Journey Beyond Normality, Chapman & Hall/CRC, Boca Raton, FL, 003, pp [6] C. Roberts, A correlation model useful in the study of twins, J. Amer. Statist. Assoc [7] S. Sahu, D. Dey, M. Branco, A new class of multivariate skew distributions with applications to Bayesian regression models, Canad. J. Statist [8] S.R. Searle, Linear Models, Wiley, New York, 97.