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.

翻译：本文提出了一种检验复合零假设的新型有限样本程序。基于渐近$χ^2$近似的传统似然比检验在小样本中常表现出显著偏差。我们的方法通过提升显著性水平实现：若对零假设区域$Θ_0$中的每个$θ_t$，其对应的简单零假设$H_0: θ= θ_t$均被拒绝，则拒绝复合零假设$H_0: θ\in Θ_0$。我们推导了确定该提升水平的计算公式，使得即使在有限样本下，整体检验仍能近似维持预设的显著性水平。传统似然比检验仅适用于由等式约束定义的零假设区域（即形成无边界流形的情形），而本文方法可扩展至同时包含等式与不等式约束的零假设。此外，该方法能处理由多个子区域并集构成的零假设，并适用于含冗余参数的模型。通过多个非标准复合零假设的示例，我们数值验证了所提检验方法能实现精确推断，其实际显著性水平与名义水平之间的差距在大小样本下均保持较小。