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Distribution of Matrix Quadratic Forms Under Skew-Normal Settings

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Distribution of Matrix Quadratic Forms Under Skew-Normal Settings Journal of Multivariate Analysis 131 (2014) 229–239 Contents lists available at ScienceDirect Journal of Multivariate Analysis journal homepage: www.elsevier.com/locate/jmva Distribution of matrix quadratic forms under skew-normal settings Rendao Ye a, Tonghui Wang b,c,∗, Arjun K. Gupta d a College of Economics, Hangzhou Dianzi University, China b Innovation Experimental College, Northwest A and F University, China c Department of Mathematical Sciences, New Mexico State University, USA d Department of Mathematics and Statistics, Bowling Green State University, USA article info a b s t r a c t Article history: For a class of skew-normal matrix distributions, the density function, moment generating Received 9 September 2013 function and independence conditions are obtained. The noncentral skew Wishart distri- Available online 15 July 2014 bution is defined, and the necessary and sufficient conditions under which a quadratic form is noncentral skew Wishart distributed random matrix are established. A new version of AMS subject classifications: Cochran's theorem is given. For illustration, our main results are applied to two examples. primary 62H10 ' 2014 Elsevier Inc. All rights reserved. secondary 62E17 Keywords: Skew-normal matrix distribution Noncentral skew Wishart distribution Generalized noncentral skew Wishart distribution Moment generating function Cochran's theorem 1. Introduction In practical applications, the skew data sets are by no means exceptions but facts of life in many diverse fields such as economics, finance, biomedicine, environment, demography, and pharmacokinetics. Usually for mathematical convenience, they are assumed to follow the normal distribution. This restrictive assumption, however, may result in not only a lack of robustness against departures from the normal distribution and but also invalid statistical inferences, especially when data are skewed (Zhang and Davidian [23]). The routine use of normality assumption has been questioned by many authors (see, e.g. [18,17,5,15,14]). Therefore, it is of both theoretical and practical importance to develop the flexible parametric classes of multivariate distributions that exhibit the skewness and kurtosis different from those for the normal distribution. In the literature, much has been studied on the class of multivariate skew-normal distributions, which includes the normal distribution and has some properties similar to the normal one while being skew (see, e.g. [3,2,7,9,10,6,8,4,20]). These contributions consider the definitions, basic properties and applications of multivariate skew-normal distribution, and discuss the distributions of its quadratic forms. However, some problems of these inferences are still open under skew-normal matrix settings, while the statistical inference methods of random matrix are broadly applied in many fields including economics, finance, biology, psychology, physics, mechanical and electrical engineering and so on (see, e.g. ∗ Corresponding author at: Department of Mathematical Sciences, New Mexico State University, USA. E-mail addresses: [email protected] (R. Ye), [email protected] (T. Wang), [email protected] (A.K. Gupta). http://dx.doi.org/10.1016/j.jmva.2014.07.001 0047-259X/' 2014 Elsevier Inc. All rights reserved. 230 R. Ye et al. / Journal of Multivariate Analysis 131 (2014) 229–239 [1,11,12]). Recently, Gupta et al. [12], in their book, provided an extension of the closed skew normal family of distributions from the vector to the matrix case using density functions, which is the generalization of many other multivariate skew normal families in the literature. Also in their book, the moment generating function of quadratic forms of matrix variate skew normal distributions is given only for the case where the location parameter is zero. It is therefore necessary to define a class of skew-normal matrix distributions without using the density function as it may not exist. Also it is useful to define the noncentral skew Wishart distributions for the case where the location parameter is not zero in order to evaluate the power functions of some test statistics. Consequently, the new version of Cochran's under skew-normal matrix settings is needed for establishing the relationships among the distributions of matrix quadratic forms. Our results extend the corresponding results given under normal matrix and skew-normal vector settings (see, e.g. [8,4,20,13,21,19,16]). This paper is organized as follows. In Section2, we discuss some properties of skew-normal matrix distribution such as its density function, moment generating function (MGF), and independence conditions. In Section3, the noncentral skew Wishart distribution is defined and several properties of a quadratic form are investigated. Section4 presents a new version of Cochran's theorem. For illustrations of our main results, two examples are given. 2. Preliminaries n 0 C − Let Mn×k be the set of all n × k matrices over the real field < and < D Mn×1. For any B 2 Mn×k, use B ; B ; B , and r.B/ to denote the transpose, the Moore–Penrose inverse, the generalized inverse, and the rank of B, respectively. For D 0 2 <k D 0 − 0 D 0 0 0 2 B .b1;:::; bn/ with bi , let PB B.B B/ B and Vec.B/ .b1;:::; bn/ . For any nonnegative definite T Mn×n and m > 0, use tr.T /; etr.T / and ρ.T / to denote the trace, the exponential trace and the spectral radius of T , respectively, m −m C and use T and T to denote the mth nonnegative definite roots of T and T , respectively. For B 2 Mm×n; C 2 Mn×p and 0 0 D 2 Mp×q, use B ⊗ C to denote the Kronecker product of B and C; Vec.BCD/ D .B ⊗ D /Vec.C/ and .B ⊗ D /.C/ D BCD. Use E and F to denote certain p- and k-dimensional inner product spaces over the real field <, use L.E; F/ to denote the vector space of all linear maps of E into F, and L.E; F/ will be equipped with the trace inner product: hB; Ci D tr.B0C/ for all B; C 2 L.E; F/. SNn.µ; Σ; α/ denotes n-dimensional multivariate skew-normal distribution, with location parameter µ, scale parameter matrix Σ and skewness parameter α. This definition can be extended to cases where the random element is an n × p matrix. Definition 2.1. The n × p random matrix Y is said to have a skew-normal matrix distribution with location matrix µ, scale 0 0 matrix V ⊗ Σ, with known V and skewness parameter matrix γ ⊗ α , denoted by Y ∼ SNn×p(µ, V ⊗ Σ; γ ⊗ α /, if n p y ≡ Vec.Y / ∼ SNnp.µ; V ⊗ Σ; γ ⊗ α/, where µ 2 Mn×p; V 2 Mn×n; Σ 2 Mp×p; µ D Vec(µ), γ 2 < , and α 2 < . From Definition 2.1, we can show that the skew-normal random matrix has the following properties. 0 0 k Proposition 2.1. Let Z ∼ SNk×p.0; Ikp; 1k ⊗ α / with 1k D .1;:::; 1/ 2 < . Then (i) The density function of Z is D 0 2 f .Z/ 2φk×p.Z/Φ.1kZα/; Z Mk×p; (1) −kp=2 0 where φk×p.Z/ D .2π/ etr.−Z Z=2/ and Φ.·/ is the standard normal distribution function. (ii) The MGF of Z is 0 0 1kT α MZ .T / D 2etr.T T =2/Φ ; T 2 Mk×p: (2) .1 C kα0α/1=2 0 Proof. We know that Z ∼ SNk×p.0; Ikp; 1k ⊗α / is equivalent to z D Vec.Z/ ∼ SNkp.0; Ikp; 1k ⊗α/. Then the density function of z is 2 0 0 f .z/ D exp.−z z=2/Φ..1k ⊗ α/ z/: (3) .2π/kp=2 Replacing z with Vec.Z/ in (3), we obtain 2 0 0 f .Z/ D exp.−Vec.Z/ Vec.Z/=2/Φ..1k ⊗ α/ Vec.Z// .2π/kp=2 2 0 0 D etr.−Z Z=2/Φ.1 Zα/: .2π/kp=2 k Further, the MGF of z is ⊗ 0 0 0 .1k α/ t kp Mz.t/ D E.exp.t z// D 2 exp.t t=2/Φ ; t 2 < : (4) .1 C kα0α/1=2 Download English Version: https://daneshyari.com/en/article/1145533 Download Persian Version: https://daneshyari.com/article/1145533 Daneshyari.com.
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