SOME ALTERNATIVES TO WALD CONFIDENCE-INTERVAL AND TEST

The problem considered is that of inference for the variance ratio lambda = sigma-s2/sigma-e2 in mixed linear models of the form y = CHI-beta + Zs + e, where beta is a column vector of unknown parameters and s and e are statistically independent, multivariate-normal random vectors, with E(s) = 0, var(s) = sigma-s2I, E(e) = 0, and var(e) = sigma-e2I. The case where there is a known upper bound on lambda is emphasized. The 100(1 - alpha)% confidence sets, corresponding to the following two-sided tests of the null hypothesis H0:lambda = lambda-0, are discussed and compared: (1) a size-alpha Wald's test and (2) the test that rejects H0 whenever H0 is rejected by the most-powerful size-alpha-1 invariant test of H0 versus the alternative lambda = lambda-u* or by the most-powerful size-alpha-2 invariant test of H0 versus the alternative lambda = lambda-l* (alpha-1 + alpha-2 = alpha, lambda-l* < lambda-0 < lambda-u*). If lambda-u* and lambda-l* are close to lambda-0, the latter test is essentially equivalent to a two-sided version of the locally-most-powerful invariant test.