Bayesian Nonparametrics, Robustness and Frechet Classes

Connections between Bayesian nonparametric inference and the classes of prob-ability measures with known marginals (Frechet classes) are investigated. Inparticular, a class of Dirichlet processes with parameters proportional to prob-ability measures in a Frechet class will be considered.Relations between two statistical characters are then studied, by presentinga new, Bayesian interpretation of the Kolmogorov-Smirnov test statistics, bylooking for upper and lower bounds on Bayes estimators, under squared lossfunction, of some indices, like covariance and concordance, and by consideringpositive dependence and \optimal" estimators of distribution functions accord-ing to some criteria (e.g. posterior regret). It should be pointed out that, ingeneral, bounds are found when considering Dirichlet processes whose parame-ter is either one of the prior Frechet bounds or one obtained by updating themafter observing a sample.Properties of some classes of probability measures are investigated too,along with their relations. In general, computing upper and lower bounds in asuitable larger class turns out to be easier than in the true class, whereas it isshown that the bounds on the two classes coincide.It is proved that, given a probability measure chosen from a Dirichlet pro-i ABSTRACT(Statistics)BAYESIAN NONPARAMETRICS, ROBUSTNESSAND FR ECHET CLASSESbyFabrizio RuggeriInstitute of Statistics and Decision SciencesDuke UniversityDate:Approved:Dr. Michael Lavine, SupervisorDr. Peter MuellerDr. Ken ReckhowDr. Brani VidakovicDr. Robert WolpertAn abstract of a dissertation submitted in partialful llment of the requirements for the degreeof Doctor of Philosophy in the Institute ofStatistics and Decision Sciences in the Graduate School ofDuke University1994(Statistics)BAYESIAN NONPARAMETRICS, ROBUSTNESSAND FR ECHET CLASSESbyFabrizio RuggeriInstitute of Statistics and Decision SciencesDuke UniversityDate:Approved:Dr. Michael Lavine, SupervisorDr. Peter MuellerDr. Ken ReckhowDr. Brani VidakovicDr. Robert WolpertAn abstract of a dissertation submitted in partialful llment of the requirements for the degreeof Doctor of Philosophy in the Institute ofStatistics and Decision Sciences in the Graduate School ofDuke University1994 BAYESIAN NONPARAMETRICS, ROBUSTNESSAND FR ECHET CLASSESbyFabrizio RuggeriInstitute of Statistics and Decision SciencesDuke UniversityDate:Approved:Dr. Michael Lavine, SupervisorDr. Peter MuellerDr. Ken ReckhowDr. Brani VidakovicDr. Robert WolpertDissertation submitted in partial ful llment of therequirements for the degree of Doctor of Philosophyin the Institute of Statistics and Decision Sciencesin the Graduate School ofDuke University1994 Copyright c1995 by Fabrizio RuggeriAll rights reserved