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Über dieses Buch

This book presents a systematic and comprehensive treatment of various prior processes that have been developed over the last four decades in order to deal with the Bayesian approach to solving some nonparametric inference problems. Applications of these priors in various estimation problems are presented. Starting with the famous Dirichlet process and its variants, the first part describes processes neutral to the right, gamma and extended gamma, beta and beta-Stacy, tail free and Polya tree, one and two parameter Poisson-Dirichlet, the Chinese Restaurant and Indian Buffet processes, etc., and discusses their interconnection. In addition, several new processes that have appeared in the literature in recent years and which are off-shoots of the Dirichlet process are described briefly. The second part contains the Bayesian solutions to certain estimation problems pertaining to the distribution function and its functional based on complete data. Because of the conjugacy property of some of these processes, the resulting solutions are mostly in closed form. The third part treats similar problems but based on right censored data. Other applications are also included. A comprehensive list of references is provided in order to help readers explore further on their own.



Chapter 1. Prior Processes

This chapter is devoted to introducing various prior processes, their formulation, properties, inter-relationships, and their relative strengths and weaknesses. The sequencing of presentation of these priors reflects mostly the order in which they were discovered and developed. The Dirichlet process and its immediate generalizations—Dirichlet Invariant and Mixtures of Dirichlet—are presented first. The neutral to the right processes and the processes with independent increments, which form the basis for many other processes, are discussed next. They are key in the development of processes that include beta, gamma and extended gamma processes, proposed primarily to address specific applications in the reliability theory, are presented next. Beta-Stacy process which extends the Dirichlet process is discussed thereafter. Following that, tailfree and Polya tree processes are presented which are especially convenient for estimating density functions, and to place greater weights, where it is deemed appropriate, by selecting suitable partitions in developing the prior. In order to extend the nonparametric Bayesian analysis to covariate data, numerous extensions are proposed. They have origin in the Ferguson-Sethuraman infinite sum representation in which the weights are constructed by a stick-breaking construction. They are collectively called here as Ferguson-Sethuraman processes and include dependent and spatial Dirichlet processes, Pitman-Yor process, Chinese restaurant and Indian buffet processes, etc. They all are included in this chapter.
Eswar G. Phadia

Chapter 2. Inference Based on Complete Data

This chapter contains various applications of prior processes discussed in the previous chapter in solving some inferential problems from a Bayesian point of view. It covers multitude of fields such as, estimation, hypothesis testing, empirical Bayes, density estimation, bioassay, etc. They are grouped according to the inferential task they signify. However, a bulk of the space is devoted to the Bayesian estimation of the distribution function, and its functional, with respect to different priors, and some common features are discussed. This is followed by confidence bands, two-sample problems, a regression problem, and some interesting additional applications are also mentioned. Finally, a decision theoretic approach to testing a statistical hypothesis regarding an unknown distribution function is indicated.
Eswar G. Phadia

Chapter 3. Inference Based on Incomplete Data

In Chap. 2, the applications were based on samples with complete data. In contrast, this chapter is devoted to presenting inferential procedures based on (mostly right) censored data. Heavy emphasis is given to the estimation of survival function since it plays an important role in the survival data analysis. Estimation procedures based on different priors and under various sampling schemes are discussed. Estimation of hazard rates and cumulative hazard functions is also included. This is followed by other examples which include estimation procedures in certain stochastic process models, Markov Chains, and competing risks models. Finally, estimation of the survival function in presence of covariates is presented.
Eswar G. Phadia


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