Estimation of Distribution Overlap of Urn Models

TL;DR

This paper introduces a new estimator for measuring the dissimilarity between two discrete distributions, demonstrating its statistical properties and applying it to microbial data to distinguish environments.

Contribution

It proposes a U-statistic based estimator for dissimilarity probability, proving its unbiasedness, convergence, and normality under certain conditions.

Findings

Estimator is a U-statistic with minimum variance.

Converges uniformly in probability to the true dissimilarity.

Applied successfully to microbial 16S rRNA data.

Abstract

A classical problem in statistics is estimating the expected coverage of a sample, which has had applications in gene expression, microbial ecology, optimization, and even numismatics. Here we consider a related extension of this problem to random samples of two discrete distributions. Specifically, we estimate what we call the dissimilarity probability of a sample, i.e., the probability of a draw from one distribution not being observed in k draws from another distribution. We show our estimator of dissimilarity to be a U-statistic and a uniformly minimum variance unbiased estimator of dissimilarity over the largest appropriate range of k. Furthermore, despite the non-Markovian nature of our estimator when applied sequentially over k, we show it converges uniformly in probability to the dissimilarity parameter, and we present criteria when it is approximately normally distributed and admits a consistent jackknife estimator of its variance. As proof of concept, we analyze V35 16S rRNA data to discern between various microbial environments. Other potential applications concern any situation where dissimilarity of two discrete distributions may be of interest. For instance, in SELEX experiments, each urn could represent a random RNA pool and each draw a possible solution to a particular binding site problem over that pool. The dissimilarity of these pools is then related to the probability of finding binding site solutions in one pool that are absent in the other.