Asymptotic mixed normality of maximum likelihood estimator for Ewens--Pitman partition

TL;DR

This paper studies the asymptotic behavior of the maximum likelihood estimator for the Ewens--Pitman partition, revealing its convergence to a mixed normal distribution and proposing methods for inference on the parameter.

Contribution

It establishes the asymptotic mixed normality of the MLE for the Ewens--Pitman partition and introduces a normalization technique to improve inference accuracy.

Findings

MLE of α is n^{α/2}-consistent

Converges to a variance mixture of normal distributions

Constructs an approximate confidence interval for α

Abstract

This paper investigates the asymptotic properties of parameter estimation for the Ewens--Pitman partition with parameters $0<\alpha<1$ and $\theta>-\alpha$. Especially, we show that the maximum likelihood estimator (MLE) of $\alpha$ is $n^{\alpha/2}$-consistent and converges to a variance mixture of normal distributions, where the variance is governed by the Mittag-Leffler distribution. Moreover, we show that a proper normalization involving a random statistic eliminates the randomness in the variance. Building on this result, we construct an approximate confidence interval for $\alpha$. Our proof relies on a stable martingale central limit theorem, which is of independent interest.