On a Theorem of Bahadur and Goodman
Erich L. Lehmann · 2011
Introduction. This note is concerned with the problem of selecting the best one (or any other specified number) of several populations. It is restricted to the symmetric case where typically the observations consist of samples of equal size from the different populations. For certain families of distributions, Bahadur (1950) and Bahadur and Goodman (1952) have proved that the natural selection procedure uniformly minimizes the risk among all symmetric procedures for a large class of loss functions. In Section 2 we give an alternative proof of this theorem, and in Section 3 show that the theorem implies many other optimum properties including one obtained in a different manner by Hall (1959).