We derive the exact asymptotic distribution of the maximum likelihood estimator $(\hat{\alpha}_n, \hat{\theta}_n)$ of $(\alpha, \theta)$ for the Ewens--Pitman partition in the regime of $0<\alpha<1$ and $\theta>-\alpha$: we show that $\hat{\alpha}_n$ is $n^{\alpha/2}$-consistent and converges to a variance mixture of normal distributions, i.e., $\hat{\alpha}_n$ is asymptotically mixed normal, while $\hat{\theta}_n$ is not consistent and converges to a transformation of the generalized Mittag-Leffler distribution. As an application, we derive a confidence interval of $\alpha$ and propose a hypothesis testing of sparsity for network data. In our proof, we define an empirical measure induced by the Ewens--Pitman partition and prove a suitable convergence of the measure in some test functions, aiming to derive asymptotic behavior of the log likelihood.

翻译：我们推导了在$0<\alpha<1$且$\theta>-\alpha$条件下，Ewens--Pitman划分中参数$(\alpha, \theta)$的最大似然估计$(\hat{\alpha}_n, \hat{\theta}_n)$的精确渐近分布：证明了$\hat{\alpha}_n$具有$n^{\alpha/2}$一致性，并收敛于正态分布的方差混合，即$\hat{\alpha}_n$是渐近混合正态的，而$\hat{\theta}_n$非一致，且收敛于广义Mittag-Leffler分布的变换。作为应用，我们推导了$\alpha$的置信区间，并提出网络数据稀疏性的假设检验。在证明中，我们定义了由Ewens--Pitman划分导出的经验测度，并证明了该测度在某些检验函数下的适当收敛性，以推导对数似然函数的渐近行为。