Exact confidence intervals and hypothesis tests for parameters of discrete distributions

TL;DR

This paper investigates the properties of exact confidence intervals and hypothesis tests for discrete distributions, highlighting issues with popular methods and establishing the optimality of fiducial intervals like Clopper-Pearson and Garwood.

Contribution

It demonstrates the lack of strict nestedness in common intervals and proves the optimality of fiducial intervals among nested options for discrete parameters.

Findings

Popular methods lack strict nestedness.

Fiducial intervals are optimal among nested intervals.

Certain methods assign different p-values to indistinguishable models.

Abstract

We study exact confidence intervals and two-sided hypothesis tests for univariate parameters of stochastically increasing discrete distributions, such as the binomial and Poisson distributions. It is shown that several popular methods for constructing short intervals lack strict nestedness, meaning that accepting a lower confidence level not always will lead to a shorter confidence interval. These intervals correspond to a class of tests that are shown to assign differing $p$-values to indistinguishable models. Finally, we show that among strictly nested intervals, fiducial intervals, including the Clopper-Pearson interval for a binomial proportion and the Garwood interval for a Poisson mean, are optimal.