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

TL;DR

This work analyzes parameter estimation for the Ewens–Pitman partition with $0<\alpha<1$ and $\theta>-\alpha$, establishing that the MLE for $\alpha$ is $n^{\alpha/2}$-consistent and converges to a variance mixture of normals driven by the generalized Mittag–Leffler variable $\mathsf{M}_{\alpha,\theta}$. A random normalization by the number of blocks $K_n$ removes the variance randomness, yielding a standard normal limit and enabling a practical confidence interval for $\alpha$. The paper also derives the asymptotic Fisher information, shows the $ heta$-estimator is non-consistent and asymptotically biased, and provides a quasi-MLE result with asymptotics, alongside an application to network sparsity. Extensions to Gibbs partitions and numerical simulations corroborate the theoretical findings, and the authors discuss a quantitative martingale CLT and future directions. Overall, the results give a precise, stable-convergence-based description of nonergodic asymptotics for MLEs in exchangeable partitions and related network models.

Abstract

This paper investigates the asymptotic properties of parameter estimation for the Ewens--Pitman partition with parameters $0<α<1$ and $θ>-α$. Especially, we show that the maximum likelihood estimator (MLE) of $α$ is $n^{α/2}$-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 $α$. Our proof relies on a stable martingale central limit theorem, which is of independent interest.

Figures (7)

• Figure 1: Asymptotic behavior of the Ewens--Pitman partition when $0<\alpha<1, \theta>-\alpha$.
• Figure 2: Plot of the Fisher Information $I_\alpha$ in \ref{['eq:sibuya_fisher']}
• Figure 3: Asymptotic orthogonality of $\alpha$ and $\theta$.
• Figure 4: Histogram of $\alpha f_\alpha^{-1}(\log {{\mathsf{M}}_{\alpha, \theta}})$ with sample size $10^6$. The solid line is the pdf of ${N}(\theta, \alpha^2 /f_\alpha'(\theta/\alpha))$, where the variance is the inverse of the asymptotic Fisher Information; that is, $\alpha^{-2} f_\alpha'(\theta/\alpha) = \lim_{n\to+\infty}\mathop{\mathrm{\mathbb{E}}}\nolimits[(\partial_{\theta} \ell_n(\alpha,\theta))^2]$.
• Figure 5: The visualization of the asymptotic mixed normality. The left figure plots the difference of CDF of $\sqrt{n^\alpha \mathop{\mathrm{\mathbb{E}}}\nolimits[{{\mathsf{M}}_{\alpha, \theta}}] I_\alpha}(\hat{\alpha_{n}}-\alpha)$ to the CDF of $N(0,1)$, while the right figure plots the difference of the CDF of $\sqrt{K_n I_\alpha}(\hat{\alpha}_n-\alpha)$ to the CDF of $N(0,1)$. Simulation setting: $\alpha=0.8$, $\theta=0$, $100000$ Monte Carlo simulations.

Theorems & Definitions (48)

• Definition 2.1: Sibuya distribution
• Definition 2.2: Generalized Mittag-Leffler distribution
• Remark 2.1
• Theorem 2.2
• proof : Sketch of proof
• Remark 2.3: pitman2006combinatorial
• Definition 2.3
• Lemma 2.4: hausler2015stable
• Proposition 3.1
• Lemma 3.2