BIAS CORRECTION FOR CENSORED DATA WITH EXPONENTIAL LIFETIMES

The analysis of censored data presents several problems including infinite maximum likelihood estimates and biased estimates. In this paper we consider modifying the score equation for the maximum likelihood estimate so that the bias is reduced, following Firth (1993). This method is considered for the case of right censored failure time data having an exponential distribution and the means of the observations are given by a log-linear model. For some situations the modified score equations can be integrated and the method is equivalent to a penalised maximum likelihood approach. We additionally show that the estimates are finite under weak conditions. A small sample study indicates that the modified estimates have good properties and have mean square error behaving like 1/n. Several problems arise with the analysis of censored data including infinite maximum likelihood estimates and biased estimates. An additional problem concerns the degrees of freedom associated with censored observations: in one extreme case observations can be practically completely missing while in the other they can be almost uncensored. Here we follow an approach which is to modify the maximum likelihood estimating equations so that the bias is reduced and in some cases finite estimates are guaranteed. For some cases the modified estimating equations can be used to obtain the estimates as maximisers of a penalised likelihood. We extend the ideas of Firth (1993) to investigate the adjustment to the likelihood score which reduces the bias of resulting estimates in a sampling theory framework. When it exists the penalty function can be interpreted as a Bayesian prior or, in a sampling framework, as available prior data.