Statistica Sinica 15(2005), 635-644

LIKELIHOOD RATIO TESTS UNDER LOCAL

ALTERNATIVES IN REGULAR SEMIPARAMETRIC

MODELS

Moulinath Banerjee

University of Michigan

Abstract: We consider the behavior of likelihood ratio statistics for testing a finite dimensional parameter, or functional of interest, under local alternative hypotheses in regular semiparametric problems. These are problems where $\sqrt{n}$-regular estimates of the parameter/functional of interest exist and, in particular, the MLE converges at $\sqrt{n}$ rate to the true value and is asymptotically normal and efficient. We show that in regular problems, the likelihood ratio statistic for testing $H_0: \theta(\psi) = \theta(\psi_0) = \theta_0$ ( where $\psi_0$ is a fixed point in the infinite-dimensional parameter space $\Psi$ and $\theta(\psi)$ is a finite-dimensional (sub)parameter or functional of interest) converges in distribution under local (contiguous) alternatives of the form $\psi_n = \psi_0 + n^{-1/2}\,h + o(n^{-1/2})$ to a non-central $\chi^2$ random variable, with non-centrality parameter involving the direction of perturbation $h$ and the efficient information matrix for $\theta$ under parameter value $\psi_0$. This conforms to what happens in the case of regular parametric models in classical statistics.

Key words and phrases: Asymptotic distribution, χ² distribution, confidence sets, contiguity, Cox model, least favorable submodels, likelihood ratio, local alternatives.