On Superiority of Estimation Regarding Pitman Closeness Criterion in General Mixed Linear Models

The general mixed linear model can be written as y = X gamma + Zu + e. In the short note, we mainly aim to apply Pitman closeness criterion to a mixed linear model. The problem of estimating the objective function as a linear combination of fixed effects and random effects, mu = L'gamma + M'u, is considered. We denote by (mu) over tilde (resp. mu*) the best linear unbiased estimate when gamma is known ( resp. unknown) of mu and by (gamma) over cap the ordinary least squares solution of gamma. Two problems are handled in sequence. Firstly, we prove that (mu) over tilde and mu* are superior over (mu) over cap (sic) L'(gamma) over cap with respect to Pitman closeness criterion by using the median unbiasedness techniques. After that, the main results are applied to Bayes arguments.