CHAPTER 12

STEIN ESTIMATION

Outline

12.1 Class of Estimators

12.2 Preliminaries and Some Theorems

12.3 Superiority Conditions

12.4 Problems

Stein (1956) showed that when the dimension of parameter space is greater than 2 (p ≥ 3), the best invariant and minimax estimator of the mean of a normal population is inadmissible. The approach by Stein (1956), which is the shrinkage approach, incorporates both uncertain prior information (on the parameter of interest) and the sample observations (from the underlying statistical distribution). To be more specific, comparing to what was proposed in previous chapters, Stein (1981) established conditions under which a more general class of shrinkage estimators dominated the usual minimax estimator under normal theory. See also the papers of James and Stein (1961), Casella (1990), Brandwein and Strawderman (1991), Ouassou and Strawderman (2002), and Xu and Izmirlian (2006), and the monographs of Lehmann and Casella (1998) and Saleh (2006) for more information. In this chapter, we consider a general class of Stein-type shrinkage estimators under the multivariate t-model.

12.1 Class of Estimators

In this chapter, we will be discussing on a more general class of Stein-type shrinkage estimators, namely the Baranchik-type (due to Baranchik, 1970) class of estimators (BTEs). In the following, we propose our study in two categories. Because the purpose of this chapter is only to consider the performance of BTE from a classical viewpoint, in the first ...

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