On the Notion of Random Intensity

Yashin, A.I. (1985). On the Notion of Random Intensity. IIASA Working Paper. IIASA, Laxenburg, Austria: WP-85-013

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Abstract

The problems of explaining the observed trends in mortality, morbidity and other kinds of individuals' transitions generated the numerous attempts of incorporating the covariates into the survival models. First models use the deterministic constant factors as explanatory variables. Gradually it became clear that the random and dynamic nature of the covariates should also be taken into account. This understanding has led to the fact that the notion of random intensity became widely used in the analysis of the asymptotic properties of the maximum likelihood and Coxregression estimators.

Having the clear intuitive sense the notion of random intensity can be introduced in different ways. The traditional way is to define the intensity in terms of probability distributions of the failure time. Another way appeals to the martingale theory and defines the intensity in terms of the predictable process, called "compensator". For the deterministic rates and simple cases of stochastic intensities there are already results that establish a one-to-one correspondence between two definitions. The correspondence is reached by the probabilistic representation results for compensator. Martingale theory guarantees the existence of the predictable compensator in more general cases. However the results on the probabilistic representation similar to simple cases are still unknown. Meanwhile such representation is crucial, for instance, in the analysis of the relations between the duration of the life cycle of some unit and stochastically changing influential variables. This paper shows the result of such representation for some particular case. The generalization on the more general situations is straightforward. The consideration will use some basic notions of a "general theory of processes".

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