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Improving on the James-Stein positive-part estimator. (English) Zbl 0820.62051
Summary: The purpose of this paper is to give an explicit estimator dominating the positive-part James-Stein rule. The James-Stein estimator improves on the “usual” estimator \(X\) of a multivariate normal mean vector \(\theta\) if the dimension \(p\) of the problem is at least 3. It has been known since at least 1964 [A. Baranchik, Tech. Rep. 51, Dpt. Stat., Stanford Univ. (1964)] that the positive-part version of this estimator improves on the James-Stein estimator. L. D. Brown’s [Ann. Math. Stat. 42, 855-903 (1971; Zbl 0246.62016)] results imply that the positive-part version is itself inadmissible although this result was assumed to be true much earlier.
Explicit improvements, however, have not previously been found; indeed, results of M. E. Bock [Statistical decision theory and related topics IV, Pap. 4th Purdue Symp., West Lafayette/ Indiana 1986, Vol. 1, 281-297 (1988; Zbl 0732.62050)] and of L. D. Brown [ibid., 299-324 (1988; Zbl 0711.62007)] imply that no estimator dominating the positive- part estimator exists whose unbiased estimator of risk is uniformly smaller than that of the positive-part estimator.

MSC:
62H12 Estimation in multivariate analysis
62C99 Statistical decision theory
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