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

Joonha Park, Ming Wang

Abstract:

We propose a novel finite-sample procedure for testing composite null hy-
potheses. Traditional likelihood ratio tests based on asymptotic χ2 approx-
imations often exhibit substantial bias in small samples. Our procedure re-
jects the composite null hypothesis H0 : θ ∈ Θ0 if the simple null hypothesis
H0 : θ= θt is rejected for every θt in the null region Θ0, 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 demon-
strate numerically that the proposed test achieves accurate inference, exhibit-
ing only a small gap between the actual and nominal significance levels for
both small and large samples.