Uniformly consistently estimating the proportion of false null hypotheses for composite null hypotheses via Lebesgue-Stieltjes integral equations

Abstract

This is the second part of the series on constructing uniformly consistent estimators of the proportion of false null hypotheses via solutions to Lebesgue-Stieltjes integral equations. We consider estimating the proportion of random variables for two types of composite null hypotheses: (i) their means or medians belonging to a non-empty, bounded interval; (ii) their means or medians belonging to an unbounded interval that is not the whole real line. For each type of composite null hypotheses, uniform consistent estimators of the proportion of false null hypotheses are constructed respectively for random variable that follow Type I location-shift family of distributions and for random variables whose distributions form continuous natural exponential families with separable moments. Further, uniformly consistent estimators of the proportion induced by a function of bounded variation on a non-empty, bounded interval are provided for the two types of random variables mentioned earlier. For each proposed estimator, its uniform consistency class and speed of convergence are provided under independence.