The Annals of Statistics

A Survey of Product-Integration with a View Toward Application in Survival Analysis

Richard D. Gill and Soren Johansen

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The correspondence between a survival function and its hazard or failure-rate is a central idea in survival analysis and in the theory of counting processes. This correspondence is shown to be a special case of a more general correspondence between multiplicative and additive matrix-valued measures on the real line. Additive integration of the survival function produces the hazard, while the multiplicative integral, or so-called product-integral, of the hazard yields the survival function. The easy generalization to the matrix case (noncommutative multiplication) allows an elegant and completely parallel treatment of intensity measures of Markov processes, with many possible applications in multistate survival models. However, the difficulties and multiplicity of theories of product-integration in multivariate time explain why so many different multivariate product-limit estimators exist. We give a complete and elementary treatment of the basic theory of the product-integral $\pi(1 + dX)$ together with a discussion of some of its applications. New results are given on the compact differentiability of the product-integral, to be used along with the functional $\delta$-method for getting large-sample results for product-limit estimators.

Article information

Ann. Statist., Volume 18, Number 4 (1990), 1501-1555.

First available in Project Euclid: 12 April 2007

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier


Primary: 60J27: Continuous-time Markov processes on discrete state spaces
Secondary: 62G05: Estimation 60H20: Stochastic integral equations 45D05: Volterra integral equations [See also 34A12]

Multiplicative integral Volterra integral equation Kolmogorov forward and backward equations Duhamel equation Doleans-Dade exponential semimartingale compact (Hadamard) differentiability Markov process intensity measure branching processes product-limit (Kaplan-Meier) estimator


Gill, Richard D.; Johansen, Soren. A Survey of Product-Integration with a View Toward Application in Survival Analysis. Ann. Statist. 18 (1990), no. 4, 1501--1555. doi:10.1214/aos/1176347865.

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