Optimization results for a generalized coupon collector problem

TL;DR

This paper analyzes a generalized coupon collector problem with a null coupon, proving that almost uniform distributions minimize collection time and certain boundary distributions maximize it, with applications in computer science.

Contribution

It introduces new stochastic bounds for collection times under arbitrary distributions, including a null coupon, and identifies extremal distributions within a class.

Findings

Almost uniform distribution minimizes collection time.

Boundary distributions maximize collection time.

Results have practical applications in computer science.

Abstract

We study in this paper a generalized coupon collector problem, which consists in analyzing the time needed to collect a given number of distinct coupons that are drawn from a set of coupons with an arbitrary probability distribution. We suppose that a special coupon called the null coupon can be drawn but never belongs to any collection. In this context, we prove that the almost uniform distribution, for which all the non-null coupons have the same drawing probability, is the distribution which stochastically minimizes the time needed to collect a fixed number of distinct coupons. Moreover, we show that in a given closed subset of probability distributions, the distribution with all its entries, but one, equal to the smallest possible value is the one, which stochastically maximizes the time needed to collect a fixed number of distinct coupons. An computer science application shows the utility of these results.