Marginal survival

Abstract

In this chapter we examine in some detail estimates of marginal survival. Some assumptions on the censoring mechanism are needed in order to make progress. Attention is paid to the exponential and piecewise exponential models. The exponential model, fully characterized by its mean, can appear over restrictive. However, via the probability integral transform and empirical estimates of marginal survival considered in this chapter, it can be used in more general situations. The piecewise exponential is seen, in some sense, to lie somewhere between the simple exponential and the empirical estimate. Particular attention is paid to empirical processes and how the Kaplan-Meier estimator, very commonly employed in survival-type problems, can be seen to be a natural generalization of the empirical distribution function. In the presence of parametric assumptions it is also straightforward to derive suitable estimating equations. The equations for the exponential model are particularly simple.