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Uniformly consistent proportion estimation for composite hypotheses via integral equations: "the case of location-shift families"

Xiongzhi Chen

TL;DR

The paper develops uniformly consistent, p-value free estimators for the proportion of false nulls under composite nulls in Type I location-shift families by solving Lebesgue-Stieltjes integral equations with discriminant functions and matching kernels. It provides two main constructions: a bounded-null estimator and a one-sided-null estimator, plus extensions to functionals of the null with bounded variation, together with explicit convergence speeds and uniform consistency classes. Theoretical results are complemented by concentration bounds and a simulation study comparing against MR/Storey methods, illustrating robust performance in dense regimes and clarifying numerical limitations in sparse settings. The approach leverages harmonic analysis, Dirichlet integrals, and Fourier-analytic tools to deliver a flexible, non-p-value-based framework for multiple testing with composite nulls, with broad potential applications and extensions to dependent data and more general null algebras.

Abstract

We consider estimating the proportion of random variables for two types of composite null hypotheses: (i) the means or medians of the random variables belonging to a non-empty, bounded interval; (ii) the means or medians of the random variables belonging to an unbounded interval that is not the whole real line. For each type of composite null hypotheses, uniformly consistent estimators of the proportion of false null hypotheses are constructed for random variables whose distributions are members of a Type I location-shift family. Further, uniformly consistent estimators of certain functions of a bounded null on the means or medians are provided for the random variables mentioned earlier; these functions are continuous and of bounded variation. The estimators are constructed via solutions to Lebesgue-Stieltjes integral equations and harmonic analysis, do not rely on a concept of p-value, and have various applications.

Uniformly consistent proportion estimation for composite hypotheses via integral equations: "the case of location-shift families"

TL;DR

The paper develops uniformly consistent, p-value free estimators for the proportion of false nulls under composite nulls in Type I location-shift families by solving Lebesgue-Stieltjes integral equations with discriminant functions and matching kernels. It provides two main constructions: a bounded-null estimator and a one-sided-null estimator, plus extensions to functionals of the null with bounded variation, together with explicit convergence speeds and uniform consistency classes. Theoretical results are complemented by concentration bounds and a simulation study comparing against MR/Storey methods, illustrating robust performance in dense regimes and clarifying numerical limitations in sparse settings. The approach leverages harmonic analysis, Dirichlet integrals, and Fourier-analytic tools to deliver a flexible, non-p-value-based framework for multiple testing with composite nulls, with broad potential applications and extensions to dependent data and more general null algebras.

Abstract

We consider estimating the proportion of random variables for two types of composite null hypotheses: (i) the means or medians of the random variables belonging to a non-empty, bounded interval; (ii) the means or medians of the random variables belonging to an unbounded interval that is not the whole real line. For each type of composite null hypotheses, uniformly consistent estimators of the proportion of false null hypotheses are constructed for random variables whose distributions are members of a Type I location-shift family. Further, uniformly consistent estimators of certain functions of a bounded null on the means or medians are provided for the random variables mentioned earlier; these functions are continuous and of bounded variation. The estimators are constructed via solutions to Lebesgue-Stieltjes integral equations and harmonic analysis, do not rely on a concept of p-value, and have various applications.

Paper Structure

This paper contains 49 sections, 15 theorems, 364 equations, 2 figures, 3 tables.

Table of Contents

  1. Introduction
  2. The strategy via solutions to Lebesgue-Stieltjes integral equations
  3. Main contributions and summary of results
  4. Notations and conventions
  5. Organization of article
  6. Preliminaries
  7. Type I location-shift family
  8. Constructions of proportion estimators for a point null
  9. Overview and illustration of constructions
  10. Constructions for Type I location-shift families
  11. The case of a bounded null
  12. The case of a one-sided null
  13. Additional examples of Type I location-shift families
  14. Extension of constructions for a bounded null
  15. Simulation study
  16. ...and 34 more sections

Key Result

Theorem 1

Let $\omega$ be an even, bounded, probability density function on $\left[ -1,1\right]$ and $\mathcal{F}$ be a Type I location-shift family. For $\mu^{\prime}\in U$, define and let Then For the point null $\Theta_{0}=\left\{ \mu_{0}\right\}$ with $\mu_{0}\in U$, $\left( \psi,K\right) =\left( \psi_{1,0}\left( t,\mu;\mu_{0}\right) ,K_{1,0}\left( t,x;\mu_{0}\right) \right)$. In particular, $

Figures (2)

  • Figure 1: We have set $t=20$ in each discriminant function $\psi\left( t,\mu\right)$, and $t=10$ in each matching function $K\left( t,x\right)$ since the magnitude of $K$ is quite large when $t$ is large. For $t=20$, $\psi\left( t,\mu\right)$ already approximates relatively well $1_{\Theta_{0}}\left( \mu\right)$ for each of the three types of nulls.
  • Figure 2: Boxplot of the excess $\tilde{\delta}_{m}$ (on the vertical axis) of an estimator $\hat{\pi}_{1,m}$ of ${\pi}_{1,m}$ in panel (a) as $\tilde{\delta}_{m}=\hat{\pi}_{1,m}\pi_{1,m}^{-1}-1$ (or an estimator $\hat{\pi}_{0,m}$ of $\tilde{\pi}_{0,m}$ in panel (b) as $\tilde{\delta}_{m}=\hat{\pi}_{0,m}\tilde{\pi}_{0,m}^{-1}-1$). The thick horizontal line and the diamond in each boxplot are respectively the mean and standard deviation of $\tilde{\delta}_{m}$, and the dotted horizontal line is the reference for $\tilde{\delta}_{m}=0$. Panel (a) is for Scenarios 1 and 2, and Panel (b) for Scenario 3, all described in \ref{['simDesign']}. All estimators were applied to Gaussian families, and the proposed estimator is referred to as "New". For for each $m$ in each case of $\pi_{1,m}$ for a one-side null, there are three boxplots, where the leftmost is for the "HD" estimator, the middle for the "MR" estimator, and the rightmost for "New". No simulation was done for the "MR" or "HD" estimator for a bounded null or estimating average, truncated 2-norm.

Theorems & Definitions (25)

  • Definition 2.1
  • Theorem 1
  • Theorem 2
  • Definition 3.1
  • Theorem 3
  • Theorem 4
  • Example 1
  • Theorem 5
  • Example 2
  • Example 3
  • ...and 15 more