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

Risk Classification 113
account for overdispersion in the Poisson part, generalizations of the model are possible and
include the Zero-Inflated Negative Binomial (ZINB) distribution. See Yip & Yau (2005) for
an application to insurance claim count data.
Other than the zero-inflated models, parametric methods such as the mixture of
distributions can be used to model the claim frequency distribution with extra zeros.
Hürlimann (1990) discussed the use of several pseudo compound Poisson distributions in
modelling the claim count data. To test for a Poisson mixture, a test statistic was proposed
by Carrière (1993a).
2.10.4 Fixed Versus Random Effects
The mixed Poisson distribution is often used to account for unknown characteristics of the
driver, influencing the number of accidents reported to the company. When panel data are
available, these hidden features can alternatively be captured by an individual heterogeneity
term that is constant over time (the standard reference for panel data is Hsiao (2003); the
particular case of count variables is treated in Cameron & Trivedi (1998)). Boucher &
Denuit (2006) compared the two approaches with emphasis on the actual meaning of the
estimated parameters in a mixed Poisson regression when random effects and covariates are
correlated. In such a case, parameter estimates should be seen as the apparent effects of the
covariates on the frequency. Keeping this in mind allows for a better understanding of the
resulting price list.
The results obtained by Boucher & Denuit (2006) legitimate the use of random effects
models even if there exists a correlation between the regressors and the heterogeneity. The
parameter estimates do not identify the impact of these regressors on the premium but only
the apparent effects. Since this is usually the focus for the actuary in ratemaking, there
is no problem with this interpretation. However, such a correlation clearly indicates that a
correction should be done to obtain a more accurate model. In particular, the apparent high
risk of young drivers should deserve some attention. The analysis conducted by Boucher
& Denuit (2006) shows that the fixed effects are very heterogeneous for these individuals.
Instead of penalizing these insureds in the a priori ratemaking, an appropriate bonus-malus
scheme could be designed. Merit rating systems improve the fairness of the tariff in that
respect. We will come back to this issue in Chapter 8.
2.10.5 Hurdle Models
Boucher, Denuit & Guillén (2006) presented and compared different risk classification
models for the annual number of claims reported to the insurer. Generalized heterogeneous,
zero-inflated, hurdle and compound frequency models are applied to a sample of an
automobile portfolio of a major company operating in Spain.
The hurdle models are widely used in connection with health care demands. An application
to credit scoring is proposed in Dionne, Artis & Guillén (1996). With health care demand,
it is generally accepted that the demand for certain types of health care services depends on
two processes: the decisions of the individual and those of the health care provider. See, e.g.
Pohlmeier & Ulrich (1995) or Santos Silva & Windmeijer (2001). The hurdle model
also possesses a natural interpretation for the number of reported claims. A reason for the
good fit of the zero-inflated models is certainly the reluctance of some insureds to report their