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

Abstract This paper investigates the asymptotic properties of parameter estimation for the Ewens–Pitman partition with parameters $0\lt\alpha\lt1$ and $\theta\gt-\alpha$ . Specifically, 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.