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-regular estimates of the parameter/functional of interest exist and, in particular, the MLE converges at
rate to the true value and is asymptotically normal and efficient. We show that in regular problems, the likelihood ratio statistic for testing
( where
is a fixed point in the infinite-dimensional parameter space
and
is a finite-dimensional (sub)parameter or functional of interest) converges in distribution under local (contiguous) alternatives of the form
to a non-central
random variable, with non-centrality parameter involving the direction of perturbation
and the efficient information matrix for
under parameter value
. 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.