Likelihood Ratio Tests and Inequality Constraints

Abstract

In likelihood ratio tests involving inequality-constrained hypotheses, the Neyman-Pearson test based on the least favourable parameter value in a compound null hypothesis can be extremely conservative. The ordinary parametric bootstrap is generally inconsistent and usually too liberal. Two methods of correcting the inconsistency of the parametric bootstrap are proposed: shrinking the constraint set toward the maximum likelihood estimate and superefficient estimation of the active set of constraints. Optimal shrinkage adjustment can be determined using bootstrap calibration. These methods are compared with the double bootstrap, the subsampling bootstrap, Bayes factors, and Bayesian P-values. The Bayesian methods are also too liberal if diffuse priors are used.