A CURIOUS EXAMPLE INVOLVING THE LIKELIHOOD RATIO TEST AGAINST ONE-SIDED HYPOTHESES

An example is presented in which the following curious phenomenon is o bserved. Let X similar to N(mu, Omega), where X = (X-1, X-2), mu = (mu (1), mu(2)); Omega(11) = Omega(22) = 1, and Omega(12) = Omega(21) = .9 0. For a random sample from N(mu, Omega) suppose that the sample mean (x) over bar = (-3, -2); thus every observed value of X-1 and X-2 Can be negative. Then, for a suitable hypothesis testing problem with mu(1 ) = 0 being the null hypothesis and <(X)over bar (1)> being the test s tatistic, one would accept that mu(1) < 0; and similarly, one would ac cept that mu(2) < 0. However, the likelihood ratio test of H-0: mu = 0 against H-1: mu greater than or equal to 0 and mu not equal 0, would reject H-0 and accept H-1. We do recognize that the hypothesis H-1: mu greater than or equal to 0 and mu not equal 0 does not allow negative values for mu(1) or mu(2). Nevertheless, the phenomenon is curious.