Asymptotic Analysis of Parameter Estimation for Ewens–Pitman Partition

07/05/2022

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by   Takuya Koriyama, et al.

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