Nonnegative Minimum Biased Quadratic Estimation in Mixed Linear Models

The problem of nonnegative quadratic estimation of a parametric function γ(β, σ)=β′Fβ+∑ri=1 fiσ2i in a general mixed linear model M{y, Xβ, V(σ)=∑ri=1 σ2iVi} is discussed. Necessary and sufficient conditions are given for y′A0y to be a minimum biased estimator for γ. It is shown how to formulate the problem of finding a nonnegative minimium biased estimator of γ 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 β2 and for σ21 are presented in explicit forms.