Annals of Statistics

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

Nils Lid Hjort

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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.

Article information

Ann. Statist., Volume 18, Number 3 (1990), 1259-1294.

First available in Project Euclid: 12 April 2007

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Primary: 62C10: Bayesian problems; characterization of Bayes procedures
Secondary: 60G57: Random measures

Beta processes censoring Cox regression cumulative hazard Levy process nonparametric Bayes time-discrete time-inhomogeneous


Hjort, Nils Lid. Nonparametric Bayes Estimators Based on Beta Processes in Models for Life History Data. Ann. Statist. 18 (1990), no. 3, 1259--1294. doi:10.1214/aos/1176347749.

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