Asymptotic Inference for Exchangeable Gibbs Partitions

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 ${\widehat{\alpha }}_n$ (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 ${\widehat{\alpha }}_n$, we construct an estimator ${\widehat{\mathsf{p}}}_n$ and derive the limit distributions of the $f$-divergence ${\mathsf{D}}_f({\widehat{\mathsf{p}}}_n\mathrm{\mid }\mathrm{\mid }{\mathsf{p}}_n)$ 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.