Nonnegative Minimum Biased Quadratic Estimation in the Linear Regression Models

In the paper the problem of nonnegative estimation of β′Hβ + hσ2 in the linear model E(y) = Xβ, Var(y)= σ2I is discussed. Here H is a nonnegative definite matrix while h is a nonnegative scalar. An iterative procedure for the nonnegative minimum biased quadratic estimator is described. Moreover, in the case that H and X′X commute, an explicit formula for this estimator is given. Admissibility of the estimator is proved. The results are applied to nonnegative estimation of the total mean squared error of a linear biased estimator.