On admissibility among affine sets of linear estimators

Graduation date: 1986We describe a general finite-dimensional inner product space setting for studying the characterization of admissible linear estimators. We extend the results of LaMotte (1982) and derive necessary and sufficient conditions for an estimator to be admissible among an arbitrary affine set of linear estimators when they are compared using quadratic risk in a linear model with general mean and variance-covariance structure. The results are shown to be applicable to linear estimation of vector-valued parametric functions compared according to total mean squared error. We also present containment results that provide a partial description of the admissible estimators for a problem. Additional results are obtained in specific vector space settings for problems where estimators are compared according to total mean squared error. We explore the admissibility of transformations of admissible estimators. We show for a linear model with general variance-covariance structure that, if H'Y is admissible among all linear (unbiased) estimators for Δ'β and D is full column-rank, then DH'Y is admissible among all linear (unbiased) estimators for DΔ'β. Admissibility is explored in models where the mean is assumed functionally independent from the variance-covariance parameters. For models with a one-dimensional design matrix, we show that under mild assumptions the admissible linear estimators consist of the scalar multiples of the unbiased admissible linear estimators, where the scalar is between 0 and 1