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

112 Actuarial Modelling of Claim Counts
Risk classification techniques for claim counts have been the topic of many papers
appearing in the actuarial literature. Early references include ter Berg (1980b) and
Albrecht (1983a,b,c). Dionne & Vanasse (1989, 1992) used a Negative Binomial
regression model, while Dean, Lawless & Willmot (1989) used a Poisson-Inverse
Gaussian distribution to fit the number of claims. Cummins, Dionne, McDonnald &
Pritchett (1990) applied the GB2 family of distributions in modelling claim counts.
Ter Berg (1996) considered the Generalized Poisson distribution of Consul (1990) and
incorporated explanatory variables with the help of a loglinear model. There are now several
textbooks devoted to the statistical analysis of count data. Let us mention Cameron &
Trivedi (1998) and Winkelmann (2003). Before the Poisson regression became popular
among actuaries, claims data were often analysed using logistic regression; see, e.g.,
Beirlant, Derveaux, De Meyer, Goovaerts, Labies & Maenhoudt (1991).
Separate analyses are usually conducted for claim frequencies and costs, including
expenses, to arrive at a pure premium. With the noticeable exception of Jorgensen
& Paes de Souza (1994), all the actuarial analyses of the pure premium so far have
examined frequencies and severities separately. This approach is particularly relevant in
motor insurance, where the risk factors influencing the two components of the pure premium
are usually different.
2.10.2 Nonlinear Effects
GLMs however only deal with categorical risk factors in an efficient way. The main drawback
of GLMs is that covariate effects are modelled in the form of a linear predictor. GLMs are
too restrictive if nonlinear effects of continuous covariates are present. Continuous covariates
can efficiently enter GLMs only if they are suitably transformed to reflect their true effect
on the score scale. However, it is not always clear how the variables should be transformed.
It has been common practice in insurance companies to model possibly nonlinear effects
of a covariate by polynomials. However, it is well known to statisticians that polynomials
are often not flexible enough to capture the variability of the data particularly when the
polynomial degree is small (see, e.g., Fahrmeir & Tutz (2001), Chapter 5). For larger
degrees the flexibility of polynomials increases but at the cost of possibly high variability
of resulting estimates particularly at the left and right extreme values of the covariate.
A more flexible approach for modelling nonlinear effects can be based on piecewise
polynomials. More specifically, the unknown functions are approximated by polynomial
splines, which may be regarded as piecewise polynomials with additional regularity
conditions (see, e.g., de Boor, 1978). We refer the interested reader to Denuit & Lang
(2004) for an overview of the existing approaches.
2.10.3 Zero-Inflated Models
Insurance data usually include a relatively large number of zeros (no claim). Deductibles
and no claim discounts increase the proportion of zeros, since small claims are not reported
by insured drivers. Zero-inflated models, including the Zero-Inflated Poisson (ZIP) model,
account for this phenomenon.
ZIP models can be considered as a mixture of a zero point mass and a Poisson distribution
and were first used to study soldering defects on print wiring boards (Lambert, 1992). To