On Negative Mass Assigned by the Bivariate Kaplan-Meier Estimator

Abstract

Conditions under which the bivariate Kaplan-Meier estimate of Dabrowska is not a proper survival function are given. All points assigned negative mass are identified under the assumption that the observations follow an absolutely continuous distribution. The number of points assigned negative mass increases as $n^2$ and the total amount of negative mass does not disappear as $n \rightarrow \infty$, where $n$ is the sample size. A simulation study is reported which shows that large amounts of negative mass are assigned by the estimator, amounts ranging from 0.3 to 0.6 for a sample of size 50 over a variety of parameters for bivariate exponential distributions.