Journal of Statistical Planning and Inference 136 (2006) 2297 – 2308 www.elsevier.com/locate/jspi Bayesian bivariate survival estimation J.K. Ghosh, N.L. Hjort, C. Messan, R.V. Ramamoorthi ∗ Department of Statistics and Probability, Michigan State University,Wells Hall, East Lansing, MI 48824-1027, USA Available online 6 September 2005 Abstract There is no easy extension of the Kaplan–Meier and Nelson–Aalen estimators to the bivariate case, and estimating bivariate survival distributions nonparametrically is associated with various nontrivial problems. The Dabrowska estimator will, for example, associate negative mass to some subsets. Bayesian methods hold some promise as they will avoid the negative mass problem, but are also prone to difficulties. We simplify and extend an example by Pruitt to show that the posterior distribution from a Dirichlet process prior is inconsistent. We construct a different nonparametric prior via Beta processes and provide an updating scheme that utilizes only the most relevant parts of the likelihood, and show that this leads to a consistent estimator. © 2005 Elsevier B.V.All rights reserved. Keywords: Bayesian nonparametrics; Beta processes; Bivariate survival; Inconsistency 1. Introduction Bivariate survival or waiting times to occurrence of some event has several interesting applica- tions. For example, a husband and his wife may both be exposed to a common risk and one might want to know if males and females have the same survival distribution. Another situation might be the study of effects of active and passive smoking when only one of the spouses smokes. In the univariate case, the most popular nonparametric estimator is the Kaplan–Meier one. The bivariate case remains surprisingly difficult in spite of a lot of work. Dabrowska (1988) constructs a bivariate analog of the Kaplan–Meier estimate which is consistent but is not a proper survival distribution in that it assigns negative mass to some events. The same is true of an earlier estimate of Langberg and Shaked (1982). Dabrowska (1988) mentions another estimate due to Bickel which is consistent and avoids negative masses but does not seem to utilize all of the data. Pruitt (1991) shows mathematically that the negative mass assigned by Dabrowska’s estimator can be ∗ Corresponding author. E-mail address: ramamoorthi@stt.msu.edu (R.V. Ramamoorthi). 0378-3758/$ - see front matter © 2005 Elsevier B.V. All rights reserved. doi:10.1016/j.jspi.2005.08.023