A rigorous lower confidence bound for the expectation of a positive random variable

TL;DR

This paper introduces a method to construct finite-sample, rigorous lower confidence bounds for the expectation of positive random variables using confidence regions for the cdf, with applications to controversial data sets.

Contribution

It presents a novel approach for finite-sample lower confidence bounds based on confidence regions for the cdf, addressing limitations of standard methods.

Findings

Provides rigorous finite-sample lower confidence bounds for positive distributions.

Analyzes asymptotic behavior of the confidence bounds.

Applies method to a controversial empirical data set.

Abstract

Given an IID sample from a positive distribution, we provide a method for constructing rigorous finite sample lower confidence bounds for the expectation of the distribution. The method is based on constructing rigorous confidence regions for the cdf of the distribution. We provide some analysis of the asymptotical behavior of the rigorous LCBs. We apply the method to obtain an LCB for a particular, controversial, empirical data set, where the validity of standard methods has been called into question.