Actuarial Modelling of Claim Counts Risk Classification, Credibility ...

48 Actuarial Modelling of Claim Counts
In a seminal paper, Simar (1976) gave a detailed description of the nonparametric
maximum likelihood estimator of F˜,
as well as an algorithm for its computation. The
nonparametric maximum likelihood estimator has a discrete distribution, and Simar (1976)
obtained an upper bound for the size of its support.
Walhin & Paris (1999) showed that, although the nonparametric maximum likelihood
estimator is powerful for evaluation of functionals of claim counts, it is not suitable for
ratemaking, because it is purely discrete. For this reason, Denuit & Lambert (2001)
proposed a smoothed version of the nonparametric maximum likelihood estimator. This
approach is somewhat similar to the route followed by Carrière (1993b), who proposed to
smooth the Tucker-Lindsay moment estimator with a LogNormal kernel.
Young (1997) applied nonparametric density estimation techniques to estimate F˜.
Because the actuary only observes claim numbers and not the conditional mean, an estimation
of the underlying risk parameter relating to the ith policy of the portfolio is the average claim
number x i (i.e. the total number of claims generated by this policy divided by the length
of the exposure period). Therefore, given a kernel K, Young (1997) suggested estimating
dF by
d̂F˜t =
n∑
i=1
( )
w i t − xi
K
h i h i
in which h i is a positive parameter called the bandwidth and w i is a weight (taken to be the
number of years the ith policy is in force divided by the total number of policy-years for the
collective). Young (1997) suggested using the Epanechnikov kernel and determined the h i s
in order to minimize the mean integrated squared error (by reference to a Normal prior).