Niels Keiding

Two classical problems in nonparametric statistical analysis of recurrent event data are considered, both formalized within the framework of a simple, stationary renewal process.

We first consider observation around a fixed time point, i.e. we observe a backward recurrence time R and a forward recurrence time S. The nonparametric maximum likelihood estimator of the gap time distribution was derived by Cox (1969) and Vardi (1985) from the length-biased distribution of the gap time R+S. However, Winter and Földes (1988) proposed to use a product-limit estimator based on S, with delayed entry given by R. We clarify the relation of that estimator to the standard left truncation problem, see Keiding and Gill (1990).

The second observation scheme considers a stationary renewal process observed in a finite interval where the left interval does not necessarily correspond to an event. (For an application to automobile insurance, see Keiding et al. (1998).) The full likelihood function is complicated, and we survey possibilities for restricting attention to various partial likelihoods, in the nonparametric case again allowing the use of simple product-limit estimators.

This is joint work with Richard D. Gill.

**References:**

Cox, D.R. (1969). Some sampling problems in technology. In: *New developments in survey sampling* (ed. N.L. Johnson and H.Smith, jr.), Wiley, New York, pp. 506-527.

Keiding, N., Andersen, C. & Fledelius, P. (1998). The Cox regression model for claims data in non-life insurance. *ASTIN Bulletin* **28**, 95-118.

Keiding, N. & Gill, R.D. (1990). Random truncation models and Markov processes. *Ann.Statist.* **18**, 582-602.

Vardi, Y. (1985). Empirical distributions in selection bias models. *Ann.Statist.* **13,** 178-203.

Winter, B.B. & Földes, A. (1988). A product-limit estimator for use with length-biased data. *Can.J.Statist.* **16,** 337-355.

**Keywords:** Survival analysis; Cox-Vardi estimator; Nonparametric maximum likelihood; Laslett's line segment problem

**Biography:** Niels Keiding is professor of biostatistics at the Faculty of Health Sciences in the University of Copenhagen in Denmark. He has worked for many years in survival analysis. He was President of the ISI from 2005 to 2007.