Abstract

In survival studies, it is often of interest to study cure rates. Sometimes the event of interest (such as death or the occurrence of a disease) may not be experienced by individuals under study, and the cure rate is the probability of the latter eventuality. Furthermore, survival times (i.e., the time to the event of interest) may be subject to random censoring due to dropping out or late entry of individuals during the study period. Interestingly, random censoring facilitates the estimation of the cure rate.

In this research, we study cure rates for multivariate survival times under multivariate random censoring. Specifically, three topics have been studied. In the first topic of this thesis, a new non-parametric multivariate cure rate estimator, based on a multivariate Kaplan-Meier estimator, is proposed. The asymptotic normality and an estimator of asymptotic variance of this estimator are obtained. In the second topic, a non-parametric cure rate estimator in the presence of covariates is constructed via kernel smoothing. The asymptotic normality of this estimator is obtained, and the optimal choice of the bandwidth via cross-validation is discussed. In the third topic of this thesis, we develop a test for the presence of immunes, i.e., we test if the cure rate is zero against the alternative that it is positive,
under univariate random censoring. The limiting distribution of the test statistic under the null hypothesis is obtained using extreme-value theory. Theoretical results are supported by simulation studies.