Estimation and Goodness-Of-Fit in the Case of Randomly Censored Lifetime Data

Several parametric, nonparametric, and semi-parametric estimators of the distribution function of a randomly right-censored random variable are compared. A new continuous distribution function estimator for randomly censored data is developed, discussed, and compared to existing estimators. Minimum distance estimation is shown to be effective in estimating Weibull location parameters when random censoring is present. A method of estimating all 3 parameters of the 3-parameter Weibull distribution using a combination of minimum distance and maximum likelihood is also given. The mean integrated squared error is estimated for each estimator using Monte Carlo simulation and Kruskal-Wallis tests are used to discern which estimators are the best in the sense of having the smallest integrated squared error. A number of new goodness-of-fit tests for randomly censored data with a composite hypothesis are introduced. Cramer-von Mises and Anderson-Darling goodness-of-fit test statistics are modified to measure the discrepancy between the maximum likelihood estimate and the Kaplan-Meier product limit estimate of the distribution function of the random variable of interest. These modified test statistics are used to construct goodness-of-fit tests for the exponential, Weibull (shape 2), and Weibull (shape 3.5) distributions when the censoring distribution is assumed to be exponential. Percentage points are obtained via Monte Carlo simulation. Another test for the exponential with exponential censoring is constructed based on the knowledge that the minimum of exponentials is also exponential. More generally, elements of competing risks theory are used to build goodness-of-fit tests using crude lifetimes. One type of test requires a parametric fit to the crude lifetimes of both the variable of interest and the censoring random variable while the other type relies on the empirical survivor function of the crude lifetime of the censoring variable.