Product Integration

This is a brief survey of product-integration for biostatisticians. 1 Product-Integration Product-integration was introduced more than 110 years ago by the Italian mathematician Vito Volterra, as a tool in the solution of a certain class of differential equations. It was studied intensively by mathematicians for half a century, but finally the subject became unfashionable and lapsed into obscurity. That is a pity, since ideas of product-integration make a very natural appearance in survival analysis, and the development of this subject (in particular, of the Kaplan-Meier estimator) could have been a lot smoother if product-integration had been a familiar topic from the start. The Kaplan-Meier estimator is the product-integral of the Nelson-Aalen estimator of the cumulative hazard function; these two estimators bear the same relation to one another as the actual survival function and the actual cumulative hazard function. There are many other applications of product-integration in survival analysis, for instance in the study of multi-state processes (connected to the theory of Markov processes), and in the theory of partial likelihood. Ordinary integration is a generalisation of summation, and properties of integrals are often easily guessed by thinking of them as sums of very, very many terms (all or most of them being very small). Similarly, product-integration generalises the taking of products; a product integral is a product of many, many terms (all or most of them being very close to the number 1). Thinking of product-integrals in this simplistic way is actually very helpful. Properties of product-integrals are easy to guess and to understand. The theory of productintegration can be a great help in studying the statistical properties of statistical quantities which explicitly or implicitly are defined in terms of product-integrals. Before defining product-integrals in general and exhibiting some of their properties, we will discuss the relation, in survival analysis, between survival