Distribution results are studied and a monotone likelihood ratio is derived. The representations are also given for some well known distributions. 1. Introduction.
Sankhy¯ a : The Indian Journal of Statistics 1995, Volume 57, Series A, Pt. 1, pp.68–78
NORMAL MIXTURE REPRESENTATIONS OF MATRIX VARIATE ELLIPTICALLY CONTOURED DISTRIBUTIONS By A.K. GUPTA and T. VARGA Bowling Green State University SUMMARY. In this paper we study matrix variate elliptically contoured distributions that admit a normal mixture representation. Distribution results are studied and a monotone likelihood ratio is derived. The representations are also given for some well known distributions.
1.
Introduction
In recent years statisticians have paid much attention to elliptically contoured distributions, i.e. Kelker (1970), Chmielewski (1980), Cambanis, Huang and Simons (1981). Fang, Kotz and Ng (1990) summarized the most important results obtained so far on this topic. Many results on matrix variate elliptically contoured distributions can be found in Fang and Anderson (1990). Definition 1.1 The p × n random matrix X has a matrix variate elliptically contoured distribution (m.e.c. distribution) if its characteristic function has the form φx(T ) = etr(iT 0 M )ψ(tr(T 0 ΣT Φ)), with T : p × n, M : p × n, Σ : p × p, Φ : n × n, Σ ≥ O, Φ ≥ 0 and ψ : [0, ∞) → O and Φ > O. We call h the density ¡ ¢ generator of the distribution. When X is matrix variate normal, h(z) = exp − z2 . We obtain ¡the ε-contaminated ¢ ¡ ¢ matrix variate normal distribution if we take h(z) = (1 − ε) exp − z2 + σεpn − 2σz 2 . The density generator ¡ ¢ pn m 2 Γ pn+m 2 h(z) = pn ¡ ¢ ¡ ¢ pn+m z 2 1 + π2 Γ m 2 m defines the matrix variate t-distribution and for m = 1 we obtain the matrix variate Cauchy distribution. Vector variate elliptical distributions that can be written as a mixture of multivariate normal distributions, were discussed by Chu (1973), Muirhead (1982) and Fang, Kotz and Ng (1990). The purpose of the present paper is to study the matrix variate elliptically contoured distributions that are mixtures of matrix variate normal distributions. In the paper, X ∼ D means X is distributed according to the distribution D, X ≈ Y means X and Y are identically distributed. The p.d.f. of X is denoted by fD (X), if X ∼ D and ED (g(X)) denotes the expected value of g(X) if X ∼ D. For a random matrix X, Cov (X) means Cov (vec (X 0 )). 2.
Mixture by distribution function
Definition 2.1. Let M : p × n, Σ : p × p, Φ : n × n be constant matrices such that Σ > O and Φ > O. Assume G(z) is a distribution function on (0, ∞). Let X ∈
