On Likelihood Ratio Tests When Nuisance Parameters are Present Only Under the Alternative

Chong-Zhi Di, Fred Hutchinson Cancer Research Center
Kung-Yee Liang, Johns Hopkins University

Abstract

In parametric models, when one or more parameters disappear under the null hypothesis, the likelihood ratio test (LRT) statistic generally does not converge to the conventional 2 distribution. Rather, it has been shown that the limiting distribution of the LRT statistic is often equivalent to that of the supremum of a squared Gaussian process. However, the limiting distribution is analytically intractable for most of examples, and approximation or simulation based methods must be used to calculate the p values. In this article, we investigate conditions under which the asymptotic distributions have analytically tractable forms, based on the principal component decomposition of Gaussian processes. When these conditions are not satisfied, the principal component representation also motivates a simple and speedy simulation algorithm to approximate p values. The results are illustrated by Anderson’s stereotype model for ordinal data.