Asymptotic Inference for Exchangeable Gibbs Partition

We study the asymptotic properties of parameter estimation and predictive inference under the exchangeable Gibbs partition, characterized by a discount parameter α∈(0,1) and a triangular array vn,k satisfying a backward recursion. Assuming that vn,k admits a mixture representation over the Ewens--Pitman family $(α, θ)$, with $θ$ integrated by an unknown mixing distribution, we show that the (quasi) maximum likelihood estimator αₙ (QMLE) for $α$ 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 pₙ, which governs the allocation of the $(n+1)$-th item, conditional on the current partition of $[n]$. Based on the asymptotics of the QMLE αₙ, we construct an estimator p̂ₙ and derive the limit distributions of the f-divergence Df(p̂ₙ||pₙ) 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.