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

Efficiency and Bonus Hunger 233
Remark 5.3 (Tweedie Generalized Linear Models) The Tweedie distributions are a threeparameter
family. They allow for any power variance function and any power link. The
Tweedie family includes the Gaussian, Poisson, Gamma and Inverse Gaussian families as
special cases. Specifically, let i = EY i be the expectation of the ith response Y i .We
assume that
p∑
q 1
i = 0 + j x ij and VY i = q 2
i
j=1
where x i is a vector of covariates and is a vector of regression cofficients, for some , q 1
and q 2 . A value of zero for q 1 is interpreted as ln i = 0 + ∑ p
j=1 jx ij . The variance power
q 2 characterizes the distribution of the responses Y . The parameter q 2 is called the index
parameter and determines the shape of the Tweedie distribution. For various values of q 2 ,
we find the following particular cases: q 2 = 0 corresponds to the Normal distribution, q 2 = 1
corresponds to the Poisson distribution, 1 2 corresponds to stable
distributions for positive continuous data.
As stated above, for 1 0 k=1 22 cik
3 2 i 2 c ik
)