arXiv:2601.02529v2 Announce Type: replace-cross Abstract: We propose a novel finite-sample procedure for testing composite null hypotheses. Traditional likelihood ratio tests based on asymptotic $\chi^2$ approximations can exhibit substantial size distortions in small samples. Our procedure rejects the composite null hypothesis $H_0: \theta \in \Theta_0$ if the simple null hypothesis $H_{0,t}: \theta = \theta_t$ is rejected for every $\theta_t \in \Theta_0$, using a suitably inflated significance level. We derive a general formula for calibrating this inflated level based on the geometry of the null region $\Theta_0$. By employing finite-sample methods for testing the simple hypotheses $H_{0,t}: \theta = \theta_t$, the proposed pointwise testing procedure reduces the discrepancy between the actual and nominal significance levels, even in small samples. Whereas the traditional likelihood ratio test applies when the null region is defined solely by equality constraints, the proposed approach extends to null hypotheses defined by both equality and inequality constraints. We also discuss the application of the pointwise testing framework to composite null hypotheses expressed as unions of several component regions and to models involving nuisance parameters. Through several examples, we demonstrate numerically that the proposed test achieves accurate Type I error control in both small- and large-sample settings.