NONPARAMETRIC ESTIMATION OF COMPOUND DISTRIBUTIONS WITH APPLICATIONS IN INSURANCE

S. M. PITTS

Department of Statistical Science, University College London, Gower Street, London WC1E 6BT, U.K.

(Received May 6, 1993; revised September 7, 1993)

Abstract. A nonparametric estimator of the distribution function G of a random sum of independent identically distributed random variables, with distribution function F, is proposed in the case where the distribution of the number of summands is known and a random sample from F is available. This estimator is found by evaluating the functional that maps F onto G at the empirical distribution function based on the random sample. Strong consistency and asymptotic normality of the resulting estimator in a suitable function space are established using appropriate continuity and differentiability results for the functional. Bootstrap confidence bands are also obtained. Applications to the aggregate claims distribution function and to the probability of ruin in the Poisson risk model are presented.

Key words and phrases: Compound distributions, nonparametric estimation, aggregate claims, probability of ruin.