THE LIKELIHOOD RATIO TEST FOR THE PRESENCE OF IMMUNES IN A CENSORED SAMPLE

Models for populations with immune or cured individuals but with other s subject to failure are important in many areas, such as medical stat istics and criminology. One method of analysis of data from such popul ations involves estimating an immune proportion I-p and the parameter( s) of a failure distribution for those individuals subject to failure. We use the exponential distribution with parameter lambda for the lat ter and a mixture of this distribution with a mass 1 - p at infinity t o model the complete data. This paper develops the asymptotic theory o f a test for whether an immune proportion is indeed present in the pop ulation, i.e., for H-0:p = 1. This involves testing at the boundary of the parameter space for p. We use a likelihood ratio test for H-0, an d prove that minus twice the logarithm of the likelihood ratio has as an asymptotic distribution, not the chi-square distribution, but a 50- 50 mixture of a chi-square distribution with 1 degree of freedom, and a point mass at 0. The result is proved under an independent censoring assumption with very mild restrictions.