Asymptotic Analysis of Parameter Estimation for Ewens--Pitman Partition

We discuss the asymptotic analysis of parameter estimation for Ewens--Pitman Partition with parameter $(\alpha, \theta)$ when $0<\alpha<1$ and $\theta>-\alpha$. For the following cases: $\alpha$ unknown $\theta$ known, $\alpha$ known $\theta$ unknown, and both $\alpha$ and $\theta$ unknown, we derive the asymptotic law of Maximum Likelihood Estimator (MLE). We show that the MLE for $\theta$ does not have consistency, whereas the MLE for $\alpha$ is $n^{\alpha/2}$-consistent and asymptotically mixed normal. Furthermore, we propose a confidence interval for $\alpha$ from a mixing convergence of the MLE. In contrast, we propose Quasi-Maximum Likelihood Estimator (QMLE) as an estimator of $\alpha$ with $\theta$ unknown, which has the same asymptotic mixed normality as the MLE. We also derive the asymptotic error of MLE and QMLE. Finally, we compare them in terms of efficiency and coverage in numerical experiments.