Uniformly consistently estimating the proportion of false null hypotheses via Lebesgue–Stieltjes integral equations

Xiongzhi Chen

Journal of Multivariate Analysis, 2019, vol. 173, issue C, 724-744

Abstract: The proportion of false null hypotheses is a very important quantity in statistical modelling and inference based on the two-component mixture model and its extensions, and in control and estimation of the false discovery rate and false non-discovery rate. Most existing estimators of this proportion threshold p-values, deconvolve the mixture model under constraints on its components, or depend heavily on the location-shift property of distributions. Hence, they usually are not consistent, applicable to non-location-shift distributions, or applicable to discrete statistics or p-values. To eliminate these shortcomings, we construct uniformly consistent estimators of the proportion as solutions to Lebesgue–Stieltjes integral equations. In particular, we provide such estimators respectively for random variables whose distributions have Riemann–Lebesgue type characteristic functions, whose distributions form discrete natural exponential families with infinite supports, and whose distributions form natural exponential families with separable moment sequences. We provide the speed of convergence and uniform consistency class for each such estimator under independence. In addition, we provide two examples for which a consistent estimator of the proportion cannot be constructed using our techniques.

Keywords: Analytic functions; Bessel functions; Concentration inequalities; Fourier transform; Lambert W functions; Lebesgue–Stieltjes integral equations; Mellin transform; Natural exponential family; Proportion of false null hypotheses (search for similar items in EconPapers)
Date: 2019
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Citations: View citations in EconPapers (2)

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DOI: 10.1016/j.jmva.2019.06.003

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