This paper contributes to the study of the random number K_n of blocks in the random partition of {1,...,n} induced by random sampling from the celebrated two-parameter Poisson–Dirichlet process. For any α ∈ (0, 1) and θ > −α, Pitman (Combinatorial Stochastic Processes (2006) Springer, Berlin) showed that (K_n)/(n^α) → S as n → +∞, where the limiting random variable, referred to as Pitman’s α-diversity, is distributed according to a polynomially scaled Mittag–Leffler distribution function. Our main result is a Berry–Esseen theorem for Pitman’s α-diversity S. The proof relies on three intermediate novel results which are of independent interest: (i) a probabilistic representation of the distribution of K_n in terms of a compound distribution; (ii) a quantitative version of the Laplace’s approximation method for integrals; (iii) a refined quantitative bound for Poisson approximation. An application of our Berry–Esseen theorem is presented in the context of Bayesian nonparametric inference for species sampling problems, quantifying the error of a posterior approximation that has been extensively applied to infer the number of unseen species in a population.
A BERRY–ESSEEN THEOREM FOR PITMAN’S α-DIVERSITY
Dolera Emanuele;
2020-01-01
Abstract
This paper contributes to the study of the random number K_n of blocks in the random partition of {1,...,n} induced by random sampling from the celebrated two-parameter Poisson–Dirichlet process. For any α ∈ (0, 1) and θ > −α, Pitman (Combinatorial Stochastic Processes (2006) Springer, Berlin) showed that (K_n)/(n^α) → S as n → +∞, where the limiting random variable, referred to as Pitman’s α-diversity, is distributed according to a polynomially scaled Mittag–Leffler distribution function. Our main result is a Berry–Esseen theorem for Pitman’s α-diversity S. The proof relies on three intermediate novel results which are of independent interest: (i) a probabilistic representation of the distribution of K_n in terms of a compound distribution; (ii) a quantitative version of the Laplace’s approximation method for integrals; (iii) a refined quantitative bound for Poisson approximation. An application of our Berry–Esseen theorem is presented in the context of Bayesian nonparametric inference for species sampling problems, quantifying the error of a posterior approximation that has been extensively applied to infer the number of unseen species in a population.I documenti in IRIS sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.
