The estimated loss frequentist approach

Kun-Liang Lu, Purdue University

Abstract

In estimation of a p-variate normal mean with identity covariance matrix, Stein-type estimators can offer significant gains in terms of risk with respect to sum of squares error loss. Their maximum risk is still equal to p, however, which will typically be their "reported loss". The estimated loss frequentist approach considers use of data-dependent "loss estimates". Two conditions that are attractive for such an estimated loss are that it be an improved estimated loss under some scoring rule, and that it has long run frequentist validity. Estimated losses with these properties are found for several of the most important Stein-type estimators. One such estimator is a generalized Bayes estimator, and the corresponding estimated loss is its posterior expected loss; thus Bayesians and frequentists can completely agree on the analysis of this problem. It is also known (see Hwang, J. and Casella, G. (1982)) that certain confidence sets recentered at Stein-type estimators have larger coverage probability than the usual confidence ellipsoids. Again, however, the minimum coverage probability (say $1 - \alpha$) of these improved sets is identical to that of the usual sets, so that only $1 - \alpha$ can be actually reported. Data dependent estimated confidence coefficients, $1 - \{\alpha}(X),$ are found which (I) have frequentist validity, (II) are always larger than $1 - \alpha$, and (III) dominate the report of $1 - \alpha$ via quadratic scoring.

Degree

Ph.D.

Advisors

Berger, Purdue University.

Subject Area

Statistics

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