Abstract: Efron and Morris (1975) considered a James-Stein (1961) estimator to predict the batting averages of major league players after using an arc sine transformation of the batting average. In this paper, the relevant asymptotic theory assuming unequal Bernoulli trials without transformation is considered. Under quadratic loss the unrestricted, restricted, preliminary test and Stein-rule estimators are compared. It is shown that although the Stein-rule estimator dominates the unrestricted estimator uniformly, it does not dominate the preliminary test estimator except for large dimensions and a range of significance levels, while both the Stein-rule and the PTE perform well relative to the unrestricted and restricted estimators.
Key words and phrases: Preliminary test estimator, Stein-rule estimator, asymptotic distributional risk, Bernoulli model.