Abstract
This paper presents a new derivation of nonparametric distribution estimation with right-censored data. It is based on an extension of the predictive inferences to compound evidence. The estimate is recursive and exact, and no stochastic approximation is needed: it simply requires that the censored data are processed in decreasing order. Only in this case the recursion provides exact posterior predictive distributions for subsequent samples under a Dirichlet process prior. The resulting estimate is equivalent to the Susarla-VanRyzin estimator and to the beta-Stacy process.
Highlights
An important extension of Bayesian nonparametric theory is the treatment of incomplete data as they involve indirect observations or latent variables
The first completely Bayesian approach to the problem of dealing with observations censored on the right was made by Susarla and Van Ryzin [15] who used a Dirichlet process [5, 2] as a prior for the random distribution F and obtained in closed form the mean of the posterior distribution of F given the data
Blum and Susarla [3] complemented this result by showing that the conditional survival function turns out to be a Dirichlet processes mixture, as considered by Antoniak
Summary
An important extension of Bayesian nonparametric theory is the treatment of incomplete data as they involve indirect observations or latent variables. In order to escape the complication of mixtures induced by incomplete data, one can use the alternative and powerful approach of Blackwell and MacQueen [2], which have shown that the extension of the Polya updating rule to a continuous space, based only on the predictive distribution, is equivalent to Ferguson’s setting In this framework the predictive distributions Fn(·) = F0(·|Dp) are easy to calculate, and Fn is a simple recursive function of Fn−1. In the right censoring case A1 ⊂ · · · ⊂ An, we prove that no stochastic approximation is necessary, as there is a sole rational pattern which recursively produces the exact posterior predictive distributions for subsequent samples under a Dirichlet process prior. In this case F is modified proportionally to its value
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