For a multinor mal distribution with a p-dimensional mean vector and an arbitrary unknown dispersion matrix , Rao (1946, 1949) proposed two tests for the problem of testing H：, unspecified, versus H：, unspecified, where . These tests are referred to as Rao’s W-test (likelihood ratio test) and Rao’s U-test (union-intersection test), respectively. This work is inspired by the well-known work of Marden and Perlman (1980) who claimed that Hotelling’s T-test is admissible while Rao’s U-test is inadmissible. Both Rao’s U-test and Hotelling’s T-test can be constructed by applying the union-intersection principle that incorporates the information for Rao’s U-test statistic but does not incorporate it for Hotelling’s T-test statistic. Rao’s U-test is believed to exhibit some optimal properties. Rao’s U-test is shown to be admissible by fully incorporating the information , but Hotelling’s T-test is inadmissible.
By the way, if time permitting, I will also briefly talk about the following two topics (both under multinormal setup): (1) The MLE of covarinace matrix. (2) P-value, power and fiducial inference (We propose a framework to achieve the goal of reconciliation of Bayesian, frequentist (Neyman-Pearson approach) and Fisherian paradigms for the problems of testing mean against restricted alternatives (closed convex cones)).