Inference: Stochastic integrals

Abstract

Further to the background theory on counting processes, martingales and stochastic integrals, outlined in Sections 2.12 and 3.6, we note that the score statistic arising from the log partial likelihood can be seen to come under the heading of a stochastic integral. The terms of this integral are then equated with those elements composing a stochastic integral and for which we can appeal to known large sample results. In the case of the multiplicative model, replacing the unknown background cumulative hazard by the Nelson-Aalen estimate produces the same result as that obtained from use of the observed information matrix. This is not the case for the additive model. In this case the variance estimator using martingale theory is generally to be preferred over the information based estimator. Considerable flexibility results from the martingale concept of conditioning on the accumulated history. One example is that of multistate processes which are easily dealt with. A second is that of obtaining procedures which are non-parametric with respect to both time and the covariate (O'Brien 1978, O'Quigley and Prentice 1991).