Estimation Of Poisson-Dirichlet Process With Empirical Bayes

As a novel but fast growing field of statistics,Bayesian nonparametrics is an in-dispensable part of Bayes statistics and widely applied in various fields,e.g.,biology,medicine,biology,machine learning,language,etc.The priors in pure Bayesian struc-ture are presumably specified,which might be challenged as the priors could not fit the data or be assigned easily,especially the more complicated process prior in Bayesian non-paramtrics.Empirical Bayes is a powerful method to get the priors or the parameters of priors based on the data and has been widely applied in many fields in its development process of more than half a century.In this dissertation,we focus on the estimates of Dirichlet process,Poisson-Dirichlet process and their mixture models by the empirical Bayes with univariate data,multivariate data,monotone missing data.The main contents of this dissertation are organized as follows:(1)In chapter 2,kinds of estimates are generated on the basis of observations,in-cluding the naive estimate,two calibrated naive estimates,and two different types of maximum likelihood estimates stemming from distinct distributions.We explore some theoretical properties and provide explicitly detailed comparisons among these estimates,in the perspectives of bias,variance,and mean squared error.Besides,we further present the corresponding calculation algorithms and numerical simulations to illustrate our the?oretical achievements.(2)In chapter 3,we considers the estimation of the unknown parameters as well as the density of the base measure in a prior under the monotone missing data structure,which are subdivided into a number of groups.The parameters are estimated by the method of maximum likelihood estimates and a set of simulations show that the estimates perform well.(3)In chapter 4,we aim to estimate the parameters of Poisson-Dirichlet mixture model with multi-group data structure by empirical Bayes.The number of mixture com-ponents with Bayesian nonparametric process priors is not fixed in advance and it can grow as the increase of data.Empirical Bayes is the useful method to estimate the mix-ture components without information on them in advance.We give the procedure to construct smooth estimates of base distribution and estimates of other two parameters,some simulations show the performance between single group and multi-group data.Also,we applied Poisson-Dirichlet mixture models to three well known real datasets.(4)In chapter 5,we study longitudinal data with random effects of Bayesian nonpara-metric priors by empirical Bayes method.On many practical situations,different data points may be influenced by the same random quantities.The priors of Dirichlet process and Poisson-Dirichlet process are fitted these cases,though,one can't exactly know the influences which are latent.For the latent variables,it is hard to the models without further assumptions.Empirical Bayes is the useful method to estimate the mixture com-ponents without information on them in advance.We give the procedure to estimate fixed effects and the parameters of random effects based on empirical nonparametrical Bayes.Some simulations are performed with Dirichlet process and Poisson-Dirichlet process.