A novel finite-sample testing procedure for composite null hypotheses via pointwise rejection
We propose a novel finite-sample procedure for testing composite null hypotheses. Traditional likelihood ratio tests based on asymptotic $χ^2$ approximations often exhibit substantial bias in small samples. Our procedure rejects the composite null hypothesis $H_0: θ\in Θ_0$ if the simple null hypothesis $H_0: θ= θ_t$ is rejected for every $θ_t$ in the null region $Θ_0$, 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.