Bayesian nonparametric survival analysis for finite populations.

Abstract

For non-Bayesian nonparametric survival analysis, the Kaplan-Meier estimator is the standard estimator. For large samples with little censoring, the KM estimator often provides a good approximation of the true survivor function. However, for small samples, or much censoring, or complex populations, there are some serious difficulties with the KM estimator. This dissertation is on Bayesian nonparametric survival analysis for a finite population with right censored data. We provide two approaches, based on the $A\sb{n}$ and $H\sb{n}$ assumptions respectively. These two assumptions were suggested by Bruce M. Hill (1968). Assume that there are n exchangeable random observations, $X\sb1{,}\...{,}X\sb{n}$. $A\sb{n}$ assumes that the probability of ties is zero, and the next observation $X\sb{n+1}$ is equally likely to fall in each of the $n + 1$ open intervals formed by the first n observations. If there are ties in the population, we break the ties by assuming that there is a tiny difference $\epsilon$ between any two tied units, and then let $\epsilon\to 0$. In this case all population values are distinct for $\epsilon > 0$. Assume that there are N units and M distinct values in the population. $H\sb{n}$ allows for ties and treats the M distinct values in the population as M categories. If the sampling scheme is a simple random sampling without replacement, then the distribution of the n observations is a mixture of multivariate hypergeometric distributions. $H\sb{n}$ assumes that all compositions of the population (i.e., the patterns of ties in the population) with the specified M and N are regarded as equally likely, as in Bose-Einstein statistics. $H\sb{n}$ also assumes that the distinct values of the realizations of the n random variables constituting the data contain no information about their ranks in the population. Simulation results show that most of the time, $A\sb{n}$ and $H\sb{n}$ statistics are more efficient than the KM statistic, and much more efficient for the case in which there is much censoring with a mixture population. The $H\sb{n}$ statistic is slightly more efficient than the $A\sb{n}$ statistic in the cases studied here.