Asymptotic Analysis of Parameter Estimation for Ewens--Pitman Partition

We discuss the asymptotic analysis of parameter estimation for Ewens--Pitman partition of parameter $(\alpha, \theta)$ when $0<\alpha<1$ and $\theta>-\alpha$. We show that $\alpha$ and $\theta$ are asymptotically orthogonal in terms of Fisher information, and we derive the exact asymptotics of Maximum Likelihood Estimator (MLE) $(\hat{\alpha}_n, \hat{\theta}_n)$. In particular, it holds that the MLE uniquely exits with high probability, and $\hat{\alpha}_n$ is asymptotically mixed normal with convergence rate $n^{-\alpha/2}$ whereas $\hat{\theta}_n$ is not consistent and converges to a positively skewed distribution. The proof of the asymptotics of $\hat{\alpha}_n$ is based on a martingale central limit theorem for stable convergence. We also derive an approximate $95\%$ confidence interval for $\alpha$ from an extended Slutzky's lemma for stable convergence.