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

Joonha Park, Ming Wang

TL;DR

This paper addresses bias in finite-sample composite-hypothesis testing by introducing a pointwise rejection framework: reject $H_0: \theta \in \Theta_0$ only if every simple null $H_0: \theta=\theta_t$, with $\theta_t\in\Theta_0$, is rejected at an inflated level $\alpha'$. Two formulas determine $\alpha'$ for different geometries of $\Theta_0$: Case A for manifolds without boundary via $\alpha' = P[\chi^2_{d_1} > \chi^2_{1-\alpha, d_1-d_0}]$, and Case B for manifolds with boundary via $1-\alpha = \tfrac{1}{2}\{ F_{\chi^2_{d_1-d_0}}(\chi^2_{1-\alpha', d_1}) + F_{\chi^2_{d_1-d_0+1}}(\chi^2_{1-\alpha', d_1}) \}$. The method extends to nuisance parameters by testing proxy simple nulls across a grid of nuisance values and yields confidence regions for the target parameter as a union over these proxies. Applications demonstrate finite-sample accuracy for interval, union, nuisance, and constrained-null regions, with favorable size and power relative to traditional LRTs and competing universal-inference methods. The approach broadens practical finite-sample inference to nonstandard null regions and offers a pathway to nonparametric extensions via bootstrap or empirical likelihood techniques.

Abstract

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.

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

TL;DR

This paper addresses bias in finite-sample composite-hypothesis testing by introducing a pointwise rejection framework: reject only if every simple null , with , is rejected at an inflated level . Two formulas determine for different geometries of : Case A for manifolds without boundary via , and Case B for manifolds with boundary via . The method extends to nuisance parameters by testing proxy simple nulls across a grid of nuisance values and yields confidence regions for the target parameter as a union over these proxies. Applications demonstrate finite-sample accuracy for interval, union, nuisance, and constrained-null regions, with favorable size and power relative to traditional LRTs and competing universal-inference methods. The approach broadens practical finite-sample inference to nonstandard null regions and offers a pathway to nonparametric extensions via bootstrap or empirical likelihood techniques.

Abstract

We propose a novel finite-sample procedure for testing composite null hypotheses. Traditional likelihood ratio tests based on asymptotic approximations often exhibit substantial bias in small samples. Our procedure rejects the composite null hypothesis if the simple null hypothesis is rejected for every in the null region , 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.
Paper Structure (13 sections, 1 theorem, 83 equations, 6 figures, 2 tables, 2 algorithms)

This paper contains 13 sections, 1 theorem, 83 equations, 6 figures, 2 tables, 2 algorithms.

Table of Contents

  1. Introduction
  2. A Pointwise Rejection Method for Testing Composite Null Hypotheses
  3. Inference in the Presence of Nuisance Parameters
  4. Applications
  5. Testing a Null Hypothesis with $\theta \in [a,b]$
  6. Testing for Two Parameters Linked by "Or"
  7. Testing in the Presence of a Nuisance Parameter
  8. Testing Null Regions Defined by Equality and Inequality Constraints
  9. Discussion
  10. Acknowledgments
  11. Disclosure statement
  12. Data Availability Statement
  13. Justification of the Modified Significance Level (Equation 3) for Manifolds with Boundary

Key Result

Proposition 1

Consider $H_0: \theta \in \Theta_0$ versus $H_1: \theta \in \Theta_1$, where $\Theta_0$ is a lower-dimensional manifold of $\Theta$ without boundary. For a given $\alpha \in (0,1)$, define so that $c_\alpha := \exp[-\frac{1}{2}\chi^2_{1-\alpha, d_1 - d_0}]$ equals $c'_{\alpha'} := \exp[-\frac{1}{2}\chi^2_{1-\alpha', d_1}]$. Let $x$ denote the observed sample. Then, the testing procedure that reje

Figures (6)

  • Figure 1: Empirical rejection rates of $H_0: \mu \in [0,1]$ for the pointwise rejection and Bonferroni methods as a function of the true mean $\mu$.
  • Figure 2: Empirical significance levels of the pointwise rejection and Bonferroni methods for varied sample sizes $n$.
  • Figure 3: Percentage of times $H_0: \beta_1 \leq 0 \text{ or } \beta_2 \leq 0$ is rejected across varying sample sizes when the true parameter values are $(\beta_1, \beta_2) = (0,0)$, $(1,0)$, and $(1,1)$. Error bars represent $1.96$ times the standard errors, and the horizontal dotted line denotes the nominal significance level of 5%.
  • Figure 4:
  • Figure 5: Comparison of the power of the proposed method based on the finite-sample $F$-test and the traditional LRT for varying values of $\psi$ and $\phi$. Error bars represent $\pm 1.96$ times the standard error of the estimated power. Dashed horizontal lines indicate the 5% significance level.
  • ...and 1 more figures

Theorems & Definitions (1)

  • Proposition 1