NONPARAMETRIC BAYESIAN-ESTIMATION OF SURVIVAL FUNCTION UNDER RANDOM LEFT TRUNCATION

Several observational studies give rise to randomly left truncated data. In a nonparametric model for such data X denotes a variable of interest, T denotes the truncation variable and the distributions of both X and T are left unspecified. For this model, the product-limit estimator, which is also the maximum likelihood estimator of the survival curve, has been widely discussed. In this article, a nonparametric Bayes estimator of the survival function based on randomly left truncated data and Dirichlet process prior is presented. Some new results on the mixtures of Dirichlet processes in the context of truncated data are obtained. These results are then used to derive the Bayes estimator of the survival function under squared error loss. The weak convergence of the Bayes estimator is studied. An example using transfusion related AIDS data quoted in Kalbfleisch and Lawless (1989) is considered.