Exponential Mixture Models with Long-Term Survivors and Covariates

Suppose a population contains individuals who may be subject to failure with exponentially distributed failure times, or else are "immune" to failure. We do not know which individuals are immune but we can infer their presence in a data set if many of the largest failure times are censored. We also have explanatory vectors containing covariate information on each individual. Models for data with such immune or "cured" individuals are of great interest in medical and criminological statistics, for example. In this paper we provide sufficient conditions for the existence, consistency, and asymptotic normality of maximum likelihood estimators for the parameters in a useful parameterization of these models. The theory is then applied to derive the asymptotic properties of the likelihood ratio test for a difference between immune proportions in a "one-way" classification. A procedure for testing the "boundary" hypothesis, that there are in fact no immunes present in data with a one-way classification, is also discussed.