Models with a Kronecker product covariance structure: Estimation and testing

Abstract

In this article we consider a pq-dimensional random vector x distributed normally with mean vector θ and the covariance matrix Λ, assumed to be positive definite. On the basis of N independent observations on the random vector x, we wish to estimate parameters and test the hypothesis H: Λ = Ψ ⊗Σ, where Ψ = (ψ ij ) : q × q and Σ = (σ ij ) : p × p, and Λ = (ψ ij Σ), the Kronecker product of Ψ and Σ. That is instead of

More on the Kronecker Structured Covariance Matrix

“…From the likelihood function, constructed of independent observation matrices, Srivastava et al (2008) proved that the maximum likelihood estimates under the restriction ψ qq = 1, where Ψ = (ψ ij ) : q × q are found by an iterative flip-flop algorithm. Srivastava et al (2008) also showed that the likelihood equations provide unique estimators. A similar algorithm has been suggested by Mardia and Goodall (1993);Dutilleul (1999);Brown et al (2001) but without the restriction ψ qq = 1.…”

“…Srivastava et al (2008) discussed estimability of the paramters under the separability assumption. From the likelihood function, constructed of independent observation matrices, Srivastava et al (2008) proved that the maximum likelihood estimates under the restriction ψ qq = 1, where Ψ = (ψ ij ) : q × q are found by an iterative flip-flop algorithm. Srivastava et al (2008) also showed that the likelihood equations provide unique estimators.…”

“…Several authors, see for example Naik and Rao (2001); Roy and Khattree (2005a); Lu and Zimmerman (2005); Mitchell et al (2005Mitchell et al ( , 2006; Srivastava et al (2008), considered estimation and testing under the separability assumption. Srivastava et al (2008) discussed estimability of the paramters under the separability assumption.…”

“…The main goal is to extend the estimation procedure, suggested by Srivastava et al (2008), for the matrix normal distribution vecX ∼ N pq (vecM , Ψ ⊗ Σ) to the case where…”

“…Particularly, estimation of a Kronecker structured covariance matrix of order three, the so called double separable covariance matrix. The estimation procedure, suggested in this paper, is a generalization of the procedure derived by Srivastava et al (2008), for a separable covariance matrix.Furthermore, the restrictions imposed by separability and double separability are discussed. …”