On the exponential model for survival

Abstract

SUMMARY A simple sufficient condition for the existence, consistency and asymptotic normality of the regression coefficient in the exponential model for survival is given. The intuition that consistent estimation can only be guaranteed when censoring variables are not too small, depending on the values of the covariates, can be quantified in some examples with the use of this condition. The exponential model for survival time is a fundamental building block in survival analysis, yet there seems to be no easily accessible condition which guarantees existence, consistency and asymptotic normality of the estimators. Well-known heuristics, such as those of Kalbfleisch & Prentice (1980, p. 51), suggest that a sufficient condition will be uniform asymptotic negligibility of the sequence of covariate vectors by comparison with the information matrix. We show that this is indeed the case; see Condition 1 below. The simplicity of this condition encourages applications, and we demonstrate, for example, how Condition 1 reveals the role of censoring in the consistency of the estimator. Since an analogue of Condition 1 is a necessary and sufficient condition for asymptotic normality of the regression coefficients in the ordinary least-squares model (Eicker, 1965), it is probably as general as could be hoped for. Our method is to extend to the survival context the methods of Fahrmeir & Kaufmann (1985), who, in the area of generalized linear models, took advantage of the 'close to linear' structure of those models to strengthen traditional asymptotic arguments for general maximum likelihood estimators. The asymptotics of survival models have been widely studied using the theory of counting processes and martingales, and Borgan (1984) uses this theory to deal with a wide class of models with covariates. To apply his results (Borgan, 1984, p. 14), however, very technical conditions involving uniform convergence of various stochastic processes must be verified. Borgan's methods rely on a traditional approach and cannot yield simple sufficient conditions like Condition 1. We need nothing more sophisticated than Chebychev's inequality and the central limit theorem for sums of independent random variables, although we are restricted to the 'close to linear' exponential model. Another approach by Friedman (1982) requires complex assumptions