  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.
