Tests for Covariance Structure in Familial Data and Principal Component

Abstract

Let x _l,...,x _n be independently and identically distributed as N_p (μ, Σ), where N_p (μ, Σ) denotes the distribution of a p-dimensional normal random vector with mean vector μ and covariance matrix Σ = (σ_ij). In the familial data analysis, certain structure on the covariance matrix is assumed. It would thus be desirable to test whether this model is correct. Similarly, in principal component analysis, it is sometimes desirable to test if some given set of orthogonal vectors are eigenvectors of the unknown covariance matrix. In this note, we consider these two problems and give likelihood ratio tests along with their asymptotic distributions; the second problem was considered by Mallows (1961). Bootstrap methods are given when normality is suspected.