Nonparametric Bayes Estimators Based on Beta Processes in Models for Life History Data

Several authors have constructed nonparametric Bayes estimators for a cumulative distribution function based on (possibly right-censored) data. The prior distributions have, for example, been Dirichlet processes or, more generally, processes neutral to the right. The present article studies the rela...

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Bibliographic Details
Published in:The Annals of statistics 1990-09, Vol.18 (3), p.1259-1294
Main Author: Hjort, Nils Lid
Format: Article
Language:English
Subjects:
60G57
62C10
Bayes estimators
Beta processes
Censored data
censoring
Censorship
Cox regression
cumulative hazard
Decision theory
Estimators
Exact sciences and technology
Kaplan Meier estimator
Levy process
Markov chains
Martingales
Mathematics
nonparametric Bayes
Nonparametric models
Probability and statistics
Probability distributions
Regression analysis
Sciences and techniques of general use
Statistics
time-discrete
time-inhomogeneous
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Summary:Several authors have constructed nonparametric Bayes estimators for a cumulative distribution function based on (possibly right-censored) data. The prior distributions have, for example, been Dirichlet processes or, more generally, processes neutral to the right. The present article studies the related problem of finding Bayes estimators for cumulative hazard rates and related quantities, w.r.t. prior distributions that correspond to cumulative hazard rate processes with nonnegative independent increments. A particular class of prior processes, termed beta processes, is introduced and is shown to constitute a conjugate class. To arrive at these, a nonparametric time-discrete framework for survival data, which has some independent interest, is studied first. An important bonus of the approach based on cumulative hazards is that more complicated models for life history data than the simple life table situation can be treated, for example, time-inhomogeneous Markov chains. We find posterior distributions and derive Bayes estimators in such models and also present a semiparametric Bayesian analysis of the Cox regression model. The Bayes estimators are easy to interpret and easy to compute. In the limiting case of a vague prior the Bayes solution for a cumulative hazard is the Nelson-Aalen estimator and the Bayes solution for a survival probability is the Kaplan-Meier estimator.
ISSN:0090-5364
2168-8966
DOI:10.1214/aos/1176347749