Conditional Tests with an Order Restriction as a Null Hypothesis

Abstract

For the isotonic normal means problem, Bartholomew (1961) discussed a conditional likelihood-ratio test of H_O: the means are homogeneous, vs. H₁: the means satisfy the linear order. He concluded that the conditional test was substantially less powerful than the chi-bar-squared test. However, for testing H₁ vs. H₂: all alternatives, the corresponding conditional test can be more powerful than the chi-bar-square test. Moreover, the conditional test can be modified so as to be asymptotically α-similar.

These conditional tests are of particular interest in general tests of simultaneous inequality constraints on parameters of asymptotically normal distributions, for which the coefficients corresponding to the p(Q.,k)’s are difficult to obtain. In this general context, the likelihood ratio statistic is asymp totically chi-bar-squared whenever the true parameter vector lies in H₁; we outline a new proof based on Silvey’s theorem that a constrained estimate and its corresponding vector of Lagrange multipliers are asymptotically normal and independent.

This research was sponsored in part by the Office of Naval Research under ONR Contract N00014-83-K-0249.

AMS 1980 subject classifications: Primary 62F03; Secondary 62H15, 62E20.