ABSTRACT
Let xa=(xa1,…, xap)', a=1,…, N, be a sample of size N(N>p) from a p-variate normal distribution with unknown mean µ and unknown positive definite covariance matrix S. On the basis of these observations we are interested in testing the hypothesis H0: µ=µ0 against the alternatives H1: µ?µ0 where S is unknown and µ0 is specified. In the univariate case (p=1) this is a basic problem in statistics with applications in every branch of applied science, and the well-known Student t-test is its optimum solution. For the general multivariate case we shall show that a multivariate analog of Student’s t is an optimum solution. This problem is commonly known as Hotelling’s problem since Hotelling (1931) first proposed the extension of Student’s t-statistic for the two-sample multivariate problem and derived its distribution under the null hypothesis. We shall now derive the likelihood ratio test of this problem. The likelihood of the observations xa, a=1,…, N is given by
Given xa, a=1,…, N, the likelihood L is a function of µ, S, for simplicity written as L (µ, S). Let O be the parametric space of (µ, S) and let ? be the subspace of O when H0; µ=µ0 is true. Under ? the likelihood function reduces to
By Lemma 5.1.1, we obtain from (7.12)
We observed in Chapter 5 that under O, L(µ, S) is maximum when
