Inadmissibility and admissibility results for unbiased loss estimators based on gauss-markov estimators

Abstract

LetY be distributed according to ann-variate normal distribution with a meanXβ and a nonsingular covariance matrixσ 2 V, where bothX andV are known,β eR p is a parameter,σ > 0 is known or unknown. Denote $$\hat \beta = (X'V^{ - 1} X)^ - X'V^{ - 1} Y$$ and $$S^2 = (Y - X\hat \beta )'V^{ - 1} (Y - X\hat \beta )$$ . Assume thatFβ is linearly estimable. Whenσ is known, it is proved that the unbiased loss estimatorσ 2tr(F(X′V −1 X)− F′) of $$(F\hat \beta - F\beta )'(F\hat \beta - F\beta )$$ is admissible for rank (F)=k≤4 and inadmissible fork ≥ 5 with the squared error loss $$[a - (F\hat \beta - F\beta )'(F\hat \beta - F\beta )]^2$$ . Whenσ is unknown and rank (X) <n, it is established that the loss estimatorcS 2, wherec is any nonnegative constant, of $$(F\hat \beta - F\beta )'(F\hat \beta - F\beta )$$ is inadmissible and that the unbiased loss estimator tr(F(X′V −1 X)− F′) of $$\sigma ^{ - 2} (F\hat \beta - F\beta )'(F\hat \beta - F\beta )$$ is admissible fork ≤ 4, and inadmissible fork ≥ 5 with squared error loss.