Some Alternatives to Wald's Confidence Interval and Test

Abstract

Abstract The problem considered is that of inference for the variance ratio λ = σ 2 s /σ 2 e in mixed linear models of the form y = Xβ + Zs + e, where β is a column vector of unknown parameters and s and e are statistically independent, multivariate-normal random vectors, with E(s) = 0, var(s) = σ 2 s I, E(e) = 0, and var(e) = σ 2 e I. The case where there is a known upper bound on λ is emphasized. The 100(1 α α)% confidence sets, corresponding to the following two-sided tests of the null hypothesis H 0:λ = λ0, are discussed and compared: (1) a size-α Wald's test and (2) the test that rejects H 0 whenever H 0 is rejected by the most-powerful size-α1 invariant test of H 0 versus the alternative λ = λ* u or by the most-powerful size-α2 invariant test of H 0 versus the alternative λ = λ* l (αl + α 2 = α, λ * l < λ 0 < λ* u ). If λ* u and λ* l are close to λ0, the latter test is essentially equivalent to a two-sided version of the locally-most-powerful invariant test.