AN OPTIMAL NONNEGATIVE PROPERTY OF AN ESTIMATOR IN MULTIVARIATE QUASI-NORMAL LINEAR MODELS

Let Y=X₁BX'₂+U_ε be a multivariate linear model, where X₁,X₂ and U≠0 are known matrices, B is an unknown matrix and ε is a random matrix. Suppose moments of the first, second and fourth order of ε have the forms as follows \begingatheredE \varepsilon=0, E \varepsilon \varepsilon^\prime=I \otimes \Sigma, \\ C_o v \varepsilon \varepsilon^\prime=2(I \otimes \Sigma) \otimes(I \otimes \Sigma)\endgathered where ∑≥0 is an unknown matrix and UU’ is an idempotent matrix. This paper gives a necessary and sufficient condition for tr （C∑）to be the uniformly minimum variance nonnegative quadratic unbiased estimator of tr （C∑） where ∑ is the least square estimator of ∑ and C≠0 is a known nonnegative definite matrix.