On the admissibility of bounds on the mean of discrete, scalar probability distributions from an iid sample

TL;DR

This paper characterizes all admissible bounds for estimating the mean of a discrete distribution from iid samples, revealing the limitations and optimality conditions of such bounds, including known bounds like the trinomial bound.

Contribution

It provides a complete characterization of admissible bounds for discrete distributions, analyzing their properties and proving the non-existence of optimal bounds in certain cases.

Findings

Certain published bounds are admissible.

The set of all admissible bounds is fully characterized.

No optimal bounds exist for large sample spaces and sample sizes.

Abstract

We address the problem of producing a lower bound for the mean of a discrete probability distribution, with known support over a finite set of real numbers, from an iid sample of that distribution. Up to a constant, this is equivalent to bounding the mean of a multinomial distribution (with known support) from a sample of that distribution. Our main contribution is to characterize the complete set of admissible bound functions for any sample space, and to show that certain previously published bounds are admissible. We prove that the solution to each one of a set of simple-to-state optimization problems yields such an admissible bound. Single examples of such bounds, such as the trinomial bound by Miratrix and Stark [2009] have been previously published, but without an analysis of admissibility, and without a discussion of the full set of alternative admissible bounds. In addition to a variety of results about admissible bounds, we prove the non-existence of optimal bounds for sample spaces with supports of size greater than 1 and samples sizes greater than 1.