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Estimation of selected parameters. (English) Zbl 1467.62016

Summary: Modern statistical problems often involve selection of populations (or genes for example) using the observations. After selecting the populations, it is important to estimate the corresponding parameters. These quantities are called the selected parameters. Using traditional estimators, such as maximum likelihood (ML) estimator, which ignores the selection can result in a large bias. It is, however, known that the Bayes estimator that ignores the selection still works well under the assumed prior distribution. But, when the prior distribution used to derive the Bayes estimator is very different from the “true” prior, the Bayes estimator can fail. The paper aims to construct estimators for the selected parameters which are robust to prior distributions. A generalization of the multiple-shrinkage Stein type estimator proposed by E. I. George [J. Am. Stat. Assoc. 81, 437–445 (1986; Zbl 0594.62061); Ann. Stat. 14, 188–205 (1986; Zbl 0602.62041)] is proposed and is shown to have a small selection bias for estimating the selected means and have an attractive small expected mean squared error. With respect to these two criteria, the proposed estimator is generally better than ML estimator, Lindley-James-Stein (LJS) estimator and Efron-Tweedie [B. Efron, J. Am. Stat. Assoc. 106, No. 496, 1602–1614 (2011; Zbl 1234.62007)] estimator.

MSC:

62-08 Computational methods for problems pertaining to statistics
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References:

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