🤖 AI Summary
This study addresses the well-known issue that conventional likelihood ratio tests exhibit substantial size distortions under small samples when testing composite null hypotheses involving equality and inequality constraints, unions of multiple regions, or nuisance parameters, particularly due to violations of the regularity condition requiring a boundary-free manifold. To overcome these limitations, the authors propose a finite-sample testing procedure that conducts pointwise simple hypothesis tests across the entire null parameter space and applies a significance-level inflation correction to achieve overall size control. This approach accommodates generalized composite null structures beyond the scope of traditional methods. Numerical experiments demonstrate that the proposed test maintains actual size extremely close to the nominal level, with negligible distortion, in both small- and large-sample settings.
📝 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.