论文标题
经验和完整的贝叶斯估计皮特曼流程的类型
Empirical and Full Bayes estimation of the type of a Pitman-Yor process
论文作者
论文摘要
皮特曼（Pitman）过程是一个随机的离散概率分布，可以将原子用于对物种的相对丰度进行建模。该过程由类型参数$σ$索引，该$ $σ$从分布的实现中控制有限样本中不同物种的数量。从Pitman-typer $σ> 0 $的Pitman流程中的大小$ n $的随机样本将包含$ n^σ$不同的值（``物种''）的顺序。在本文中，我们考虑了经验贝叶斯和完整贝叶斯方法对类型参数的估计。我们在经常贝叶斯后部的经验贝叶斯估计量和Bernstein-von Mises定理的渐近正态性中得出，在常见的频繁设置中，观察值是来自给定的真实分布的随机样本。我们还考虑了Pitman-Yor过程的第二个参数的估计，即先前的精度。我们将结果应用于在法医统计的环境中得出似然比的极限行为。
The Pitman-Yor process is a random discrete probability distribution of which the atoms can be used to model the relative abundance of species. The process is indexed by a type parameter $σ$, which controls the number of different species in a finite sample from a realization of the distribution. A random sample of size $n$ from the Pitman-Yor process of type $σ>0$ will contain of the order $n^σ$ distinct values (``species''). In this paper we consider the estimation of the type parameter by both empirical Bayes and full Bayes methods. We derive the asymptotic normality of the empirical Bayes estimator and a Bernstein-von Mises theorem for the full Bayes posterior, in the frequentist setup that the observations are a random sample from a given true distribution. We also consider the estimation of the second parameter of the Pitman-Yor process, the prior precision. We apply our results to derive the limit behaviour of the likelihood ratio in a setting of forensic statistics.
