The maximum negative hypergeometric distribution

TL;DR

This paper introduces the maximum negative hypergeometric distribution, a finite-sample distribution modeling the minimum number of draws needed to observe a specified number of both colors in urn sampling without replacement, along with its properties and estimation methods.

Contribution

It defines and analyzes the maximum negative hypergeometric distribution, providing modes, approximations, and estimation techniques for urn contents, extending previous work on related distributions.

Findings

Derived the distribution's properties and modes.

Proposed approximation methods for practical use.

Developed estimation procedures for urn contents.

Abstract

An urn contains a known number of balls of two different colors. We describe the random variable counting the smallest number of draws needed in order to observe at least $\,c\,$ of both colors when sampling without replacement for a pre-specified value of $\,c=1,2,\ldots\,$. This distribution is the finite sample analogy to the maximum negative binomial distribution described by Zhang, Burtness, and Zelterman (2000). We describe the modes, approximating distributions, and estimation of the contents of the urn.