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Uniformly consistent proportion estimation for composite hypotheses via integral equations: "the case of Gamma random variables"

Xiongzhi Chen

TL;DR

This work develops uniformly consistent estimators for the proportion of false null hypotheses under composite nulls for Gamma-distributed variables, using a strategy based on discriminant functions and matching kernels that solve Lebesgue-Stieltjes integral equations. Key innovations include Construction I (bounded null) and Construction II (one-sided null), both leveraging Dirichlet integrals and harmonic analysis to approximate indicator sets without relying on p-values, and an extension to functionals of a bounded null with bounded variation. The Gamma family’s scaling-invariance and separable-moments enable precise control of convergence speeds and uniform consistency classes under independence, with simulations showing competitive performance against MR and Storey methods. The approach broadens applicability to adaptive FDR/FNR procedures for composite-null testing and suggests extensions to other distributions and Lie-group-valued data via harmonic-analytic techniques.

Abstract

We consider estimating the proportion of random variables for two types of composite null hypotheses: (i) the means of the random variables belonging to a non-empty, bounded interval; (ii) the means 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 the Gamma family. Further, uniformly consistent estimators of certain functions of a bounded null on the means 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.ce via mixture models, and may be used to estimate the sparsity level in high-dimensional Gaussian linear models.

Uniformly consistent proportion estimation for composite hypotheses via integral equations: "the case of Gamma random variables"

TL;DR

This work develops uniformly consistent estimators for the proportion of false null hypotheses under composite nulls for Gamma-distributed variables, using a strategy based on discriminant functions and matching kernels that solve Lebesgue-Stieltjes integral equations. Key innovations include Construction I (bounded null) and Construction II (one-sided null), both leveraging Dirichlet integrals and harmonic analysis to approximate indicator sets without relying on p-values, and an extension to functionals of a bounded null with bounded variation. The Gamma family’s scaling-invariance and separable-moments enable precise control of convergence speeds and uniform consistency classes under independence, with simulations showing competitive performance against MR and Storey methods. The approach broadens applicability to adaptive FDR/FNR procedures for composite-null testing and suggests extensions to other distributions and Lie-group-valued data via harmonic-analytic techniques.

Abstract

We consider estimating the proportion of random variables for two types of composite null hypotheses: (i) the means of the random variables belonging to a non-empty, bounded interval; (ii) the means 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 the Gamma family. Further, uniformly consistent estimators of certain functions of a bounded null on the means 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.ce via mixture models, and may be used to estimate the sparsity level in high-dimensional Gaussian linear models.

Paper Structure

This paper contains 44 sections, 16 theorems, 281 equations, 1 figure, 1 table.

Table of Contents

  1. Introduction
  2. Review of major existing works
  3. Main contributions and summary of results
  4. Notations and conventions
  5. Organization of article
  6. Preliminaries
  7. The estimation problem
  8. The strategy via solutions to Lebesgue-Stieltjes integral equations
  9. The Gamma family
  10. Constructions of proportion estimators for a point null and Dirichlet integrals
  11. An overview on the new constructions
  12. Constructions for Gamma family
  13. The case of a bounded null
  14. The case of a one-sided null
  15. Extension of constructions for a bounded null
  16. ...and 29 more sections

Key Result

theorem 1

Let $\omega$ be an even, bounded, probability density function on $\left[ -1,1\right]$. For $\theta^{\prime}\in\Theta$, define and let Then For the point null $\Theta_{0}=\left\{ \theta_{0}\right\}$ with $\theta_{0}\in\Theta$, $\left( \psi,K\right) =\left( \psi_{3,0}\left( t,\theta;\theta_{0}\right) ,K_{3,0}\left( t,x;\theta_{0}\right) \right)$ when $\mathcal{F}$ is the Gamma family. In pa

Figures (1)

  • Figure 1: Boxplot of the excess $\tilde{\delta}_{m}$ (on the vertical axis) of an estimator $\hat{\pi}_{1,m}$ of ${\pi}_{1,m}$ as $\tilde{\delta}_{m}=\hat{\pi}_{1,m}\pi_{1,m}^{-1}-1$. 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 $\tilde{\delta}_{m}=0$. An estimator with a narrower boxplot that is closer to the dotted horizontal line is better. All estimators have been applied to Gamma family. For the case of a one-side null, the right one for each pair of boxplots for each $m$ is for the proposed estimator "New" and the left one is for the "MR" estimator. No simulation was done for the "MR" estimator for a bounded null. Note that the "Method" legend for boxplots is basically invisible in the subplots since each boxplot contains observations that vary so little and are hence very narrow vertically.

Theorems & Definitions (23)

  • definition thmcounterdefinition
  • theorem 1
  • theorem 2
  • theorem 3
  • theorem 4
  • theorem 5
  • definition thmcounterdefinition
  • lemma thmcounterlemma
  • theorem 6
  • theorem 7
  • ...and 13 more