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NONPARAMETRIC ESTIMATION OF COMPOUND

DISTRIBUTIONS WITH APPLICATIONS IN INSURANCE

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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.

**Source**
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