Asymptotic Inference for Exchangeable Gibbs Partitions

Abstract

We study the asymptotic properties of parameter estimation and predictive inference under the exchangeable Gibbs partition, characterized by a discount parameter $\alpha \in (0,1)$ and a triangular array $v_{n,k}$ satisfying a backward recursion. Assuming that $v_{n,k}$ admits a mixture representation over the Ewens--Pitman family $(\alpha ,\theta )$, with $\theta$ integrated by an unknown mixing distribution, we show that the (quasi) maximum likelihood estimator ${\hat{\alpha }}_n$ (QMLE) for $\alpha$ is asymptotically mixed normal. This generalizes earlier results for the Ewens--Pitman model to a more general class. We further study the predictive task of estimating the probability simplex ${\mathsf{p}}_n$, which governs the allocation of the $(n+1)$-th item, conditional on the current partition of $[n]$. Based on the asymptotics of the QMLE ${\hat{\alpha }}_n$, we construct an estimator ${\hat{\mathsf{p}}}_n$ and derive the limit distributions of the $f$-divergence ${\mathsf{D}}_f({\hat{\mathsf{p}}}_n||{\mathsf{p}}_n)$ for general convex functions $f$, including explicit results for the TV distance and KL divergence. These results lead to asymptotically valid confidence intervals for both parameter estimation and prediction.

Accepted to Stochastic Processes and their Applications. We have added a simple formula using sin(x) for the Fisher Information of the Sibuya distribution