Bayesian bivariate survival estimation

TL;DR

This paper addresses the challenge of estimating bivariate survival distributions nonparametrically, proposing a Bayesian approach with Beta processes that ensures consistency.

Contribution

It introduces a new nonparametric Bayesian prior using Beta processes, overcoming limitations of existing methods like the Dabrowska estimator.

Findings

The Dirichlet process prior leads to inconsistent estimators.

Beta process prior yields a consistent estimator.

The proposed method improves nonparametric bivariate survival estimation.

Abstract

There is no easy extension of Kaplan-Meier and Nelson-Aalen estimators to the bivariate case, and estimating bivariate survival distributions nonparametrically is associated with various non-trivial 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, butare 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.