Hypothesis Tests with Multiple Linear Inequality Constraints

Abstract

We consider the problem of testing the null hypothesis that a multivariate normal mean vector is constrained to lie in a set satisfying multiple inequality constraints, which can also be described as testing multivariate one-sided null hypotheses. Lehmann (1952) showed that unbiasedness does not hold for such tests. The likelihood ratio test is a conventional approach for the problem. But, it is argued that this naive test tends to have very low powers on the boundary of the null parameter space and give biased results in favor of parameters of higher dimension. We give examples showing the bias of the traditional test and present a new test having much better properties. Geometrically, our null hypothesis is a polyhedral convex subset of a Euclidean space, and the alternative hypothesis is the complement of the null. The new test consists of two stages: In the first stage, a face of the null hypothesis is selected according to the value of PIC (first stage information criterion). In the second stage, the null hypothesis is rejected if the likelihood ratio statistic is greater than a critical value that depends on the face chosen in the first stage. The new test is constructed such to be pointwise asymptotically unbiased. Perlman and Wu (2006) investigated a subclass of our testing problem and proposed a new test, which was also shown much less biased than the likelihood ratio test. We also compared our results to theirs.