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.
翻译:我们讨论对Ewens-Pitman 分区参数估算的无症状分析,使用(alpha,\theta) $0 ALpha < 1美元和$\theta>-alpha美元时使用 $0 ALpha < 1美元和$\theta>-alpha美元。对于以下案例,我们讨论对Ewens-Pitman 分区参数估算的无症状分析。对于已知的美元、已知的美元、已知的美元、已知的美元和未知的美元,我们讨论对Ewens-Pitman 分区参数估算的无症状分析。我们提出最大相似模拟器(MLE)的无症状法度法度法度。我们表明,$(theta) MLE 的MLE 值与 值覆盖值并不一致,而无症状的MLE 值也是正常的。 我们的MLE 和MLA 的数值的正值的正值和正数性之间,我们也得出了正常的中间值。