A reversal phenomenon in estimation based on multiple samples from the Poisson--Dirichlet distribution

TL;DR

This paper investigates how the Fisher information comparison between two sampling methods from a Poisson-Dirichlet distribution can reverse depending on parameters, revealing complex behavior in statistical information measures.

Contribution

It uncovers a reversal phenomenon in Fisher information between two sampling schemes from the Poisson-Dirichlet distribution based on parameter regimes.

Findings

Fisher information comparison can reverse depending on parameters

Small θ favors sample (i) in Fisher information

Large θ favors sample (ii) in Fisher information

Abstract

Consider two forms of sampling from a population: (i) drawing $s$ samples of $n$ elements with replacement and (ii) drawing a single sample of $ns$ elements. In this paper, under the setting where the descending order population frequency follows the Poisson--Dirichlet distribution with parameter $\theta$, we report that the magnitude relation of the Fisher information, which sample partitions converted from samples (i) and (ii) possess, can change depending on the parameters, $n$, $s$, and $\theta$. Roughly speaking, if $\theta$ is small relative to $n$ and $s$, the Fisher information of (i) is larger than that of (ii); on the contrary, if $\theta$ is large relative to $n$ and $s$, the Fisher information of (ii) is larger than that of (i). The result represents one aspect of random distributions.