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On the admissibility of scale and quantile estimators Brewster, John Frederick

Abstract

The inadmissibility of the best affine-invariant estimators for the variance and noncentral quantiles of the normal law, when loss is squared error, has already been established. However, the proposed (minimax) alternatives to the usual (minimax, but inadmissible) estimators are themselves inadmissible. In our search for admissible alternatives in these problems, we first consider estimators which are formal Bayes within the class of scale-invariant procedures. For such estimators, we present explicit conditions for admissibility within the class of scale-invariant procedures. In the second chapter of the thesis, we consider the estimation of an arbitrary power of the scale parameter of a normal population. Under the assumption that the loss function satisfies certain reasonable conditions, an estimator is constructed which is (i) minimax, and (ii) formal Bayes within the class of scale-invariant procedures. The estimator obtained is a limit of a sequence of minimax, preliminary test estimators. Moreover, under squared error loss, and using the results of Chapter One, this estimator is shown to be scale-admissible. More generally, results are obtained for the estimation of powers of the scale parameter in the canonical form of the general linear model, and for the estimation of powers of the scale parameter of an exponential distribution with unknown location. In Chapter Three, conditions are given for the minimaxity of best invariant procedures in general location-scale problems. Finally, by combining these results with those of the preceding chapter, the usual interval estimators for the variance of a normal population are shown to be minimax, but inadmissible. Superior, minimax procedures are suggested.

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