Testing for the mean vector of a multivariate normal distribution with a possibly singular dispersion matrix and related results

Let y [approximate] p ([mu], [Sigma]) where [Sigma], possibly singluar, is unknown. We develop a test for H0: [mu] = [mu]0 against H1:[mu][not equal to][mu]0 based on a random random sample fromy as an extension of Hotelling's T2 test. We show that the same procedure can be used to compute the test statistic in both the cases of the singular and the positive definite dispersion matrices. This development involves the singular Wishart distribution. Motivated by Khatri (1968), we also obtain an expression for the density of the Wishart distribution p(k, [Sigma]) in each o the following cases: (i)[Sigma] positive definite, k < p, (ii)[Sigma] positive semidefinite of rank r (< p) and k [greater-or-equal, slanted] r, (i of rank r (< p) and k < r.