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

256 Actuarial Modelling of Claim Counts
distributions as the Normal and Gamma, the purely discrete scaled Poisson distribution, as
well as the class of mixed compound Poisson distribution with Gamma summands. The
name Tweedie has been associated with this family by Jorgensen (1987,1997) in honour
of the pioneering works by Tweedie (1984).
In nonlife ratemaking, the Tweedie model is very convenient for risk classification.
However, it does not allow the actuary to isolate the frequency part of the pure premium,
and thus does not provide the actuary with the input for the design of bonus-malus scales.
This is why in this book (which is mainly devoted to motor insurance pricing) we favoured
the separate analysis of claim costs and claim sizes. For other insurance products, where
only the total amount of claims is available for actuarial analysis, the Tweedie distribution
is an excellent candidate for loss modelling.
In the actuarial literature, Jorgensen & Paes de Souza (1994) assumed Poisson arrival
of claims and Gamma distributed costs for individual claims. These authors directly modelled
the risk or expected cost of claims per insured unit using the Tweedie Generalized Linear
Model. Smyth & Jorgensen (2002) observed that, when modelling the cost of insurance
claims, it is generally necessary to model the dispersion of the costs as well as their mean.
In order to model the dispersion, these authors used the framework of double generalized
linear models. Modelling the dispersion increases the precision of the estimated tariffs. The
use of double generalized linear models also allows the actuary to handle the case where
only the total cost of claims and not the number of claims has been recorded.
5.5.3 Large Claims
The analysis of large losses performed in this chapter is based on Cebrian, Denuit &
Lambert (2003). Large losses are modelled using the Generalized Pareto distribution, and
the main concern is to determine the threshold between small and large losses. An alternative
has been developed by Buch-Kromann (2006) based on Buch-Larsen, Nielsen, Guillén
& Bolanće (2005). This approach is based on a Champernowne distribution, corrected
with a nonparametric estimator (that is obtained by transforming the data set with the
estimated modified Champernowne distribution function and then estimating the density of
the transformed data set using the classical kernel density estimator). Based on the analysis of
a Danish data set, Buch-Kromann (2006) concluded that the Generalized Pareto approach
performs better than the Champernowne one in terms of goodness-of-fit, whereas both
methods are comparable in terms of predicting future claims.
Another approach is proposed by Cooray & Ananda (2005) who combined a LogNormal
probability density function together with a Pareto one. Specifically, these authors introduced
a two-parameter smooth continuous composite LogNormal-Pareto model that is a twoparameter
LogNormal density up to an unknown threshold value and a two-parameter Pareto
density for the remainder. Continuity and differentiability are imposed at the unknown
threshold to ensure that the resulting probability density function is smooth, reducing the
number of parameters from four to two. The resulting two-parameter probability density
function is similar in shape to the LogNormal density, yet its upper tail is thicker than the
LogNormal density (and accomodates to the large losses observed in liability insurance).
This approach clearly outperforms the one proposed in this chapter, in that all the parameters
(including the threshold) are estimated in the same model. The approaches obtained with
the methodology developed in this book can be used as starting values in the maximum