On the independence of singular multivariate skew-normal sub-vectors

Accepted Manuscript On the independence of singular multivariate skew-normal sub-vectors Phil D. Young, David J. Kahle, Dean M. Young PII: DOI: Reference:

S0167-7152(16)30203-6 http://dx.doi.org/10.1016/j.spl.2016.08.021 STAPRO 7727

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Statistics and Probability Letters

Received date: 22 June 2016 Revised date: 25 August 2016 Accepted date: 31 August 2016 Please cite this article as: Young, P.D., Kahle, D.J., Young, D.M., On the independence of singular multivariate skew-normal sub-vectors. Statistics and Probability Letters (2016), http://dx.doi.org/10.1016/j.spl.2016.08.021 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.

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On the Independence of Singular Multivariate Skew-Normal Sub-vectors Phil D. Younga,∗, David J. Kahleb , Dean M. Youngb a One

Bear Place, #98005, Department of Management and Information Systems, Baylor University, Waco, Texas 78798-7140 b One Bear Place, #97140, Department of Statistical Science, Baylor University, Waco, Texas 78798-7140

Abstract This article provides necessary and sufficient conditions that characterize independence among sub-vectors of a singular multivariate skew-normal random vector. Keywords: Affine subspace, Moore-Penrose inverse, Matrix column space, Matrix null space 1. Introduction Introduced by [10], the multivariate skew-normal (MSN) distribution, a generalization of the multivariate normal distribution, has become a useful and often applied alternative to the multivariate normal in situations where asymmetry is present [9]. As such, its theoretical properties have drawn considerable attention within the statistical literature. Although a skewed ∗ Corresponding

author Email addresses: [email protected] (Phil D. Young), [email protected] (David J. Kahle), [email protected] (Dean M. Young)

Preprint submitted to Statistics & Probability Letters

August 24, 2016

univariate normal distribution was considered by [1] and [2], the present-day univariate skew-normal distribution was developed by [6] as an extension of the normal density function and further developed by [7] to incorporate an additional shape parameter. Investigations into the MSN distribution and its applications continue to be fruitful. Recently, [15] highlighted a connection between the skew-normal distribution and independent component analysis popular in machine learning contexts, and [16] investigated the effect of (hidden) truncation on independence relationships among components of a normal vector, which might become skew normal under special circumstances. [11] provided necessary and sufficient conditions for conditional independence among components of a skew-normal vector. Many others have also contributed to the long list of MSN applications, including [5], [14], [13], [8], [20], and [4]. This paper considers the results of two recent articles on the singular multivariate skew-normal distribution (SMSN), a broad family that includes regular and singular multivariate symmetric and skew-normal distributions. In the first article, [21] derived, among other properties, necessary and sufficient conditions for independence of two SMSN components. In the second, [22] derived the densities of SMSN random vectors. Here, we use the results of the second article to provide a characterization of independence among components of an SMSN random vector that is slightly different from that of the first article. The remainder of the paper is organized as follows. In Section 2, we 2

establish basic notation, definitions, and two lemmas enabled by [22] and needed for the main result. In Section 3, we present the main result of the article. We conclude with some brief remarks in Section 4. 2. Preliminary Results 2.1. Notation We first establish notation that we use throughout the remainder of the paper. Let Rm×n represent the vector space of all m × n matrices over

the real field R, Sn the cone of all n × n real symmetric matrices, and Sn

and Sn the cones of all real symmetric nonnegative definite and positive

definite matrices, respectively, in Rn×n . For a matrix A ∈ Rm×n , let N (A) and C(A) denote the null space and column space of A, respectively. The Moore-Penrose pseudoinverse of A is denoted by A+ . Finally, letting λi (A) denote the i-th largest eigenvalue of A ∈ Sn , we define the product detr (A) =

r Y

i=1

where 1 ≤ r < n.

3

λi (A),

2.2. The Singular Multivariate Normal (SMN) and Singular Multivariate Skew-Normal (SMSN) Distributions The standard construction of the multivariate normal distribution is for one to apply affine transformations of a collection of independent standard normal random variables [18]. That is, if z = (z1 , . . . , zp )0 is a p-vector of independent and identically distributed standard normal random variables, A ∈ Rp×p , and µ ∈ Rp , then x = Az + µ has a multivariate normal distribution with mean µ and covariance matrix Σ = AA0 . In this case we write x ∼ Np (µ, Σ). If rank(Σ) = rank(A) = r < p, then Σ is said to be rank deficient, and the distribution of x is said to be singular multivariate normal (SMN) [17]. Random vectors that follow an SMN distribution do not admit densities with respect to the Lebesgue measure on Rp [12]; however, they do admit densities with respect to an r-dimensional affine subspace of Rp . Moreover, from [17], one can take this density to be the definition of the SMN family. Definition 1. A random vector x ∈ Rp is said to have an SMN distribution if its probability density function (PDF) can be written as φSp (x; µ, Σ)

=

(2π)–p/2 detr (Σ)–1/2 exp



 1 0 + – (x – µ) Σ (x – µ) , 2

(1)

p

where µ ∈ Rp and Σ ∈ S with rank(Σ) = r < p. Note that (1) is defined with respect to the Lebesgue measure on V = µ + C(Σ), an r-dimensional 4

affine subspace of Rp . Using an approach similar to that of [19], [22] demonstrated that the singular multivariate skew-normal (SMSN) distribution, constructed via an affine transformation of a basic skew-normal random vector, also admits a density with respect to the Lebesgue measure on a lower dimensional affine subspace of Rp . Moreover, the following density can be taken to be the definition of the SMN family of random variables. Definition 2. A random vector x ∈ Rp is said to follow an SMSN distri-

bution with skewness parameter γ ∈ Rp , written x ∼ SNp (µ, Σ, δ0 , γ), if its PDF can be written as 1 φS (x; µ, Σ)Φ f(x) = Φ(δ0 ) p

δ0 + γ 0 Σ+ (x – µ) p 1 – γ 0 Σ+ γ

!

,

(2)

where φSp (·) is the SMN density given in (1), x, µ, and γ are p-vectors, p Σ ∈ S with rank(Σ) = r < p, γ ∈ C(A) such that γ 0 Σ+ γ < 1, δ0 is a real

number, and Φ(·) is the univariate standard normal cumulative distribution function (CDF). This density is also defined with respect to the Lebesgue measure on V = µ + C(Σ). In both the multivariate normal and multivariate skew-normal cases, distribution singularity corresponds to singularity in the dispersion parameter Σ and results in the concentration of the distibution entirely on an rdimensional affine subspace of Rp . In the multivariate normal case, Σ is the

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covariance matrix of the random vector. In the multivariate skew-normal case, the covariance matrix is a function of both the dispersion parameter Σ and the skewness parameter γ. Also, the distribution mean is a function of the location parmaeter µ and the skewness parameter γ. The independence of two SMSN subvectors is more complex. 2.3. Preliminary Lemmas This section provides two lemmas concerning distributions associated with components of an SMSN vector x = [x01 : x02 ]0 . The first lemma characterizes the marginal distributions of the components of SMSN vectors as SMSN subvectors. The second lemma characterizes the conditional distributions of certain SMSN subvectors, which we show does not necessarily follow an SMSN distribution. Lemma 1. Let x ∼ SNp (µ, Σ, δ0 , γ), where rank(Σ) = r < p and γ ∈ C(Σ). Partition x, µ, Σ, and γ such that   x1  x =  , x2

  µ1  µ =  , µ2

  Σ11 Σ12  Σ= , Σ012 Σ22

  γ1  and γ =   , γ2

(3)

q

where x1 , µ1 , γ1 ∈ Rq , and Σ11 ∈ S with Σ12 ∈ C(Σ11 ) and Σ22 – p–q

0 Σ+ Σ ∈ S Σ12  . If rank(Σ11 ) = q < r, then x1 ∼ SNq (µ1 , Σ11 , δ0 , γ1 ), 11 12

6

and its marginal density is  0 Σ+ (x – µ ) δ + γ 1 2 2  0 h(x1 ) = φSq (x1 ; µ1 , Σ11 )Φ  q 1 11 Φ(δ0 ) 1 – γ10 Σ+ 11 γ1 

(4)

with γ1 ∈ C(Σ11 ) and γ10 Σ+ 11 γ1 < 1. Proof. The proof is a direct consequence of the SMSN MGF given in [22].  We remark that because x1 is arbitrary, each of the elements of the lemma is also true for any subvector, including x2 . Lemma 2. Under the assumptions of Lemma 1,

g(x1 |x2 ) =

  1 1/2 ϕSm (x1 ; µ1|2 , Σ1|2 )Φ l0 + l01 (Σ1|2 )+ (x1 – µ1|2 ) , Φ(β0 )

(5)

where µ1|2 = µ1 + Σ12 Σ+ 22 (x2 – µ2 ),

(6)

0 Σ1|2 = Σ11 – Σ12 Σ+ 22 Σ12 ,

(7)

δ0 + γ20 Σ+ (x2 – µ2 ) p 22 , 1 – γ 0 Σ+ γ δ0 + γ20 Σ+ 22 (x2 – µ2 ) , q β0 = 1 – γ20 Σ+ 22 γ2 l0 =

7

(8) (9)

and

1/2

(Σ1|2 )+ (γ1 – Σ12 Σ+ 22 γ2 ) p l1 = . 1 – γ 0 Σ+ γ

(10)

Proof. Recall the formula   A11 A12  Σ+ =   A012 A22 with + + + 0 + 0 A11 = Σ+ 11 + Σ11 Σ12 (Σ22 – Σ12 Σ11 Σ12 ) Σ12 Σ11 + 0 + A12 = –Σ+ 11 Σ12 (Σ22 – Σ12 Σ11 Σ12 ) ,

and + A22 = (Σ22 – Σ012 Σ+ 11 Σ12 ) .

8

(11)

Using (2) and Lemma 1, we have

g(x1 |x2 ) =

1 ϕS (x; µ, Σ)Φ Φ(δ0 ) p



δ0√ +γ 0 Σ+ (x–µ) 1–γ 0 Σ+ γ

1 ϕS (x ; µ , Σ )Φ 22 Φ(δ0 ) p–m 2 2

1 = ϕS (x ; µ , Σ )Φ Φ(β0 ) m 1 1|2 1|2 =



δ0 +γ20 Σ+ 22 (x2 –µ2 ) q 1–γ20 Σ+ 22 γ2

1 ϕS (x ; µ , Σ ) × Φ(β0 ) m 1 1|2 1|2

!

δ0 + γ 0 Σ+ (x – µ) p 1 – γ 0 Σ+ γ

!

δ0 + γ20 Σ+ (x2 – µ2 ) (γ1 – Σ12 Σ+ 22 γ2 ) Σ+ (x – µ ) p 22 Φ + p 1|2 1|2 1 1 – γ 0 Σ+ γ 1 – γ 0 Σ+ γ   1 1/2 = ϕSm (x1 ; µ1|2 , Σ1|2 )Φ l0 + l01 (Σ1|2 )+ (x1 – µ1|2 ) , Φ(β0 ) where µ1|2 , Σ1|2 , l0 , β0 , and l1 are given in (6)–(10), respectively.

!



3. Independence of SMSN Random Sub-Vectors In Theorem 2.2 of [21], the authors proposed necessary and sufficient conditions for the independence of sub-vectors x1 and x2 of a skew-normal random vector with a singular dispersion parameter Σ. In the language of Lemma 1, the authors claimed that x1 and x2 are independent if and only if: (i) Σ12 = 0, and 1/2

1/2

(ii) Σ11 γ1 = 0 or Σ22 γ2 = 0.

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That is, they require only that Σ12 = 0 and γ1 ∈ N (Σ11 ) or that γ2 ∈ N (Σ22 ). Below, we use the results of Lemmas 1 and 2 to determine a slightly different set of necessary and sufficient conditions for the independence of two SMSN random sub-vectors that correct this claim. In particular, we show that item (ii) of Theorem 2.2 of [21] is, in fact, insufficient, and the slightly more stringent condition that γ1 = 0 or γ2 = 0 is required. Because γ1 and γ2 are the skewness parameters of the marginal densities, we demonstrate that either one or both of the random subvectors, x1 or 2, are multivariate normal and are possibly singular. The following theorem characterizes the independence of sub-vectors of a (possibly singular) multivariate skew-normal random vector in terms of its dispersion parameter Σ and skewness parameter γ. Theorem. Let x ∼ SNp (µ, Σ, δ0 , γ) where γ ∈ C(Σ), and let x, µ, Σ, q

and γ be partitioned as in (3). Also, let Σ12 ∈ S , Σ12 ∈ C(Σ11 ), and p–q

0 Σ+ Σ ∈ S Σ22 – Σ12  . Then x1 and x2 are independent if and only if 11 12

(a) Σ12 = 0, and (b) γ1 = 0 or γ2 = 0.

Proof. Let h(x1 ) be the marginal density of x1 given in Lemma 1 and g(x1 |x2 ) be the conditional density of x1 given x2 in the proof of Lemma 2.

10

First, suppose x1 is independent of x2 so that g(x1 |x2 ) = h(x1 ). From (6), (7), and (10), we must have that µ1|2 = µ1 , Σ1|2 = Σ11 , and, consequently, Σ12 ∈ N (Σ11 ). Moreover, from Lemma 2 we have that Σ12 ∈ C(Σ11 ) and, hence, Σ12 = 0 so that (a) in the theorem must follow. Moreover, from (8), (9), and (10), we must have that γ2 ∈ N (Σ22 ). However, from Lemma 1, γ2 ∈ C(Σ22 ) and, thus, γ2 = 0. Therefore, (b) in the theorem also holds. Next, assume (a) and (b). Then from (5)–(10) and some algebra, one can readily demonstrate that g(x1 |x2 ) = h(x1 ), where h(x1 ) is given in (4). 

Thus, x1 and x2 are independent. 4. Conclusion

As one can see, condition (b) in the theorem is more restrictive than condition (ii) in Theorem 2.2 of [21]. The fact that γ, the skewness parameter, must necessarily be contained in the column space of the dispersion parameter Σ yields this more restrictive condition in the theorem. In closing, we remark on a stream of research closely related to the current article. Researchers have investigated numerous generalizations of the multivariate skew-normal family. [3] provided a concise overview of these investigations while proposing a new family called the unified skew normal (SUN), which generalizes the SMSN in this work. In the same work, the authors provided the density of the full-rank and singular SUN distributions and characterized independence among components in terms of the disper11

sion and skewness parameters for a full-rank dispersion matrix. However, the proof presented for the independence of two SUN sub-vectors with a positivedefinite dispersion matrix does not hold for two SMSN components with a rank-deficient dispersion matrix.

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