A novel finite-sample testing procedure for composite null hypotheses via pointwise rejection

TL;DR

This paper introduces a finite-sample testing procedure for composite null hypotheses that maintains accurate significance levels in small samples, extending traditional likelihood ratio tests to more complex hypotheses.

Contribution

The paper presents a new finite-sample test that handles composite nulls with inequalities, unions, and nuisance parameters, improving inference accuracy over traditional methods.

Findings

Achieves accurate significance levels in small and large samples.

Extends to null hypotheses with inequalities and unions.

Demonstrates superior performance through numerical examples.

Abstract

We propose a novel finite-sample procedure for testing composite null hypotheses. Traditional likelihood ratio tests based on asymptotic $\chi^2$ approximations often exhibit substantial bias in small samples. Our procedure rejects the composite null hypothesis $H_0: \theta \in \Theta_0$ if the simple null hypothesis $H_0: \theta = \theta_t$ is rejected for every $\theta_t$ in the null region $\Theta_0$, using an inflated significance level. We derive formulas that determine this inflated level so that the overall test approximately maintains the desired significance level even with small samples. Whereas the traditional likelihood ratio test applies when the null region is defined solely by equality constraints--that is, when it forms a manifold without boundary--the proposed approach extends to null hypotheses defined by both equality and inequality constraints. In addition, it accommodates null hypotheses expressed as unions of several component regions and can be applied to models involving nuisance parameters. Through several examples featuring nonstandard composite null hypotheses, we demonstrate numerically that the proposed test achieves accurate inference, exhibiting only a small gap between the actual and nominal significance levels for both small and large samples.