A result on hypothesis testing for a multivariate normal distribution when some observations are missing

Let Ui = (Xi, Yi), i = 1, 2,..., n, be a random sample from a bivariate normal distribution with mean [mu] = ([mu]x, [mu]y) and covariance matrix . Let Xi, i = n + 1,..., N represent additional independent observations on the X population. Consider the hypothesis testing problem H0 : [mu] = 0 vs. H1 : [mu] [not equal to] 0. We prove that Hotelling's T2 test, which uses (Xi, Yi), i = 1, 2,..., n (and discards Xi, I = n + 1,..., N) is an admissible test. In addition, and from a practical point of view, the proof will enable us to identify the region of the parameter space where the T2-test cannot be beaten. A similar result is also proved for the problem of testing [mu]x - [mu]y = 0. A Bayes test and other competitors which are similar tests are discussed.