Distributions of Quadratic Forms in Linear Mixed Models with Skew-Normal Random Effects

Abstract

Distributions of Quadratic Forms in Linear Mixed Models with Skew-Normal Random Effects Tonghui Wang1 1Department of Mathematical Sciences, New Mexico State University, USA ?: For the linear mixed model with skew-normal random effects, properties of quadratic forms are discussed. The noncentral skew chi-square distribution is defined and its density function is given. The necessary and sufficient conditions under which the quadratic form is distributed as noncentral skew chi-square distribution are obtained. Also, a version of Cochran’s theorem is obtained, which modifies the result of Wang et al. (2009). Furthermore, our main results are applied to establish the exact tests for fixed effects and the variance component. For illustrations, several examples are given.