Asymptotic behavior of the number of distinct values in a sample from the geometric stick-breaking process

TL;DR

This paper analyzes the asymptotic growth of the number of distinct values in samples from the geometric stick-breaking process, revealing how different prior choices influence this behavior and providing detailed expansions for comparison.

Contribution

It investigates the effect of success probability distributions on the asymptotics of distinct values in the geometric stick-breaking process, offering new insights and detailed expansions.

Findings

Range of logarithmic behaviors identified

Two-term asymptotic expansion derived

Comparison with negative binomial-based measures included

Abstract

Discrete random probability measures are a key ingredient of Bayesian nonparametric inferential procedures. A sample generates ties with positive probability and a fundamental object of both theoretical and applied interest is the corresponding random number of distinct values. The growth rate can be determined from the rate of decay of the small frequencies implying that, when the decreasingly ordered frequencies admit a tractable form, the asymptotics of the number of distinct values can be conveniently assessed. We focus on the geometric stick-breaking process and we investigate the effect of the choice of the distribution for the success probability on the asymptotic behavior of the number of distinct values. We show that a whole range of logarithmic behaviors are obtained by appropriately tuning the prior. We also derive a two-term expansion and illustrate its use in a comparison with a larger family of discrete random probability measures having an additional parameter given by the scale of the negative binomial distribution.