Set-indexed survival analysis with generalized censoring

Abstract

This thesis focuses on the problem of survival analysis of data subject to generalized censoring by an arbitrary adapted random set. We consider the problem of estimation of both the cumulative hazard and the distribution function. A problem with observability of certain events needed to construct the Nelson-Aalen estimator for the cumulative hazard is resolved by introducing a new model based on a more general type of random set. Under the framework of this censoring scheme, we reconstruct the Nelson-Aalen estimator and propose three different estimators for the distribution function: the first two come from a reverse probability equation and the last is a path-dependent estimator. Functional central limit theorems are proven for each of these estimators which, coupled with bootstrapping methods, allow us to explore some applications, like tests for independence of the components in the bivariate case, a test of hazard rate order and copula estimation. The novelty of this work is twofold: first, the techniques developed may be applied not only to distributions on Euclidean spaces, but also to distributions on more general metric spaces, and second, the generalized censoring mechanism subsumes and extends the usual models used in multivariate survival analyis.