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Abstract
This dissertation develops a number of nonparametric inference procedures for time to
event data under right censoring. The type of event time data that we consider are broader than the usual survival time data. They include some multistage models such as the competing risk and also certain mark variables and covariates. More specifically, we address the issue of testing the equality of two or more survival distributions when the population membership information is not available for the right censored individuals. This can also be regarded as testing the independence of failure time and cause in a competing risk problem or, more generally, as testing the independence of a failure time and a mark variable. We introduce a family of weighted log-rank tests based on the concept of assigning only a fraction of a censored individual to each at risk set of failures. The joint asymptotic normality of our test statistics is explored through an asymptotic linear representation. In addition,
two resampling schemes are suggested as alternatives to the asymptotic distribution which might be more useful in practice. Another major accomplishment of this dissertation is to formulate a class of U-statistics for right censored data problems that are based on the concept of data reweighting and are valid for a kernel of arbitrary order. They are useful for asymptotically unbiased estimation for certain mean-like functionals of the failure time distributions. We obtain a martingale
representation for these statistics which leads to their asymptotic normality. The issue of efficiency gain through the use of a doubly robust version is also discussed. As a motivating application of this estimation methodology, we construct a second test statistic for the testing problem described earlier. The finite sample properties of our estimators and statistical tests are studied through extensive simulations and two well known data sets from existing literature are used for their
illustrations to real data problems.