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

64 Actuarial Modelling of Claim Counts
2.3.2 Loglinear Poisson Regression Model
In Poisson regression, we have a collection of independent Poisson counts whose means are
modelled as nonnegative functions of covariates. Specifically, let N i , i = 1 2n,bethe
number of claims reported by policyholder i and d i be the corresponding risk exposure (the
N i s are assumed to be independent). All the observable characteristics (the a priori variables
presented in Section 2.2 for Portfolio A, say) related to this policyholder are summarized
into the vector xi
T = x i1 x ip . Poisson regression is a technique analogous to linear
regression except that errors no longer follow a Normal distribution but the randomness
in the model is described by a Poisson distribution. We first assume that the conditional
expectation of N i given x i is of the form
(
)
p∑
EN i x i = d i exp 0 + j x ij i= 1 2n (2.1)
j=1
where T = 0 1 p is the vector of unknown regression coefficients. The
explanatory variables enter the model in the linear combination 0 + ∑ p
j=1 jx ij , where 0
acts as an intercept and j is the coefficient indicating the weight given to the jth covariate.
The Poisson regression model consists in stating that N i is Poisson distributed with mean
given by Expression (2.1), that is
( (
))
p∑
N i ∼ oi d i exp 0 + j x ij i= 1 2n
2.3.3 Score
The quantity
j=1
score i = 0 +
p∑
j x ij
is called the score (or linear predictor in statistics) because it allows the actuary to rank the
policyholders from the least to the most dangerous. The claim frequency for policyholder i is
d i expscore i ; its annual claim frequency being expscore i . Increasing the score thus means
that the associated average annual claim frequency increases. The use of the exponential link
function ensures the claim frequency is positive even if the score is negative.
Let us denote as ̂ 0 ̂ 1 ̂ p the estimators of the regression coefficients
0 1 p . In a statistical sense,
̂i = d i exp ( (
)
) p∑
ŝcore i = di exp ̂0 + ̂j x ij
is the predicted expected number of claims for policyholder i. Prediction in this sense does
not refer to ‘predicting the future’ (called forecasting by statisticians) but rather to guessing
the expected number of claims (i.e., the response) from the values of the regressors in an
observation taken under the same circumstances as the sample from which the regression
equation was estimated.
j=1
j=1