On asymptotically optimal tests under loss of identifiability in semiparametric models

TL;DR

This paper develops asymptotically optimal tests for hypotheses in semiparametric models with nonidentifiable parameters, extending existing parametric results to more complex models with infinite-dimensional nuisance parameters.

Contribution

It introduces exponential average tests based on integrated profile likelihood that are optimal under a weighted average power criterion, applicable to models with nonidentifiable or non-estimable parameters.

Findings

Tests are asymptotically optimal under various alternatives.

Proposed bootstrap method effectively computes critical values.

Simulation shows improved power over existing tests.

Abstract

We consider tests of hypotheses when the parameters are not identifiable under the null in semiparametric models, where regularity conditions for profile likelihood theory fail. Exponential average tests based on integrated profile likelihood are constructed and shown to be asymptotically optimal under a weighted average power criterion with respect to a prior on the nonidentifiable aspect of the model. These results extend existing results for parametric models, which involve more restrictive assumptions on the form of the alternative than do our results. Moreover, the proposed tests accommodate models with infinite dimensional nuisance parameters which either may not be identifiable or may not be estimable at the usual parametric rate. Examples include tests of the presence of a change-point in the Cox model with current status data and tests of regression parameters in odds-rate models with right censored data. Optimal tests have not previously been studied for these scenarios. We study the asymptotic distribution of the proposed tests under the null, fixed contiguous alternatives and random contiguous alternatives. We also propose a weighted bootstrap procedure for computing the critical values of the test statistics. The optimal tests perform well in simulation studies, where they may exhibit improved power over alternative tests.