Nonnegative Minimum Biased Quadratic Estimation in Mixed Linear Models

The problem of nonnegative quadratic estimation of a parametric function [gamma]([beta], [sigma])=[beta]'F[beta]+[summation operator]ri=1 fi[sigma]2i in a general mixed linear model {y, X[beta], V([sigma])=[summation operator]ri=1 [sigma]2iVi} is discussed. Necessary and sufficient conditions are given for y'A0y to be a minimum biased estimator for [gamma]. It is shown how to formulate the problem of finding a nonnegative minimium biased estimator of [gamma] as a conic optimization problem, which can be efficiently solved using convex optimization techniques. Models with two variance components are considered in detail. Some applications to one-way classification mixed models are given. For these models minimum biased estimators with minimum norms for square of expectation [beta]2 and for [sigma]21 are presented in explicit forms.