RESTRICTED RISK BAYES ESTIMATION

SHUN-YU CHEN, Purdue University

Abstract

Let X be an observation from a p-variate continuous distribution from the exponential family with density f(x(VBAR)(theta)). Under the quadratic loss L((theta),(delta)) = ((theta)-(delta))('T)Q((theta)-(delta)) we want to find an estimator which minimizes r((pi),(delta)) subject to the constraint R((theta),(delta)) - R((theta),(delta)(,0)) (LESSTHEQ) K((theta)) for all (theta) (ELEM) (THETA), where (delta)(,0) is a 'standard' estimator, K is a nonnegative function and (pi) is a given prior distribution. Using Stein's unbiased estimator of risk to yield the representation R((theta),(delta)) - R((theta),(delta)(,0)) = E ((DELTA)(gamma)(x)), where (gamma) is an appropriate function and (DELTA) a nonlinear differential operator, we actually consider the approximate restricted Bayes problem of minimizing r((pi),(delta)) subject to a constraint on (DELTA)(gamma)(x). We first obtain a characterization theorem for the approximate restricted Bayes problem. Applying this result to the problem of estimating a multivariate normal mean in the nonsymmetric case, several good estimators are proposed when (pi) is a conjugate normal prior. Examples of restricted risk Bayes problems for the one dimensional normal distribution and Gamma distribution are also discussed.

Degree

Ph.D.

Subject Area

Statistics

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