It says in the core reading that the claim amounts are modelled with a Gamma error term with a log link function. But isn't the link function for the Gamma being 1/Mu? i.e. g(Mu) = 1/Mu But the core reading says the link function is ln(y)... Can someone please explain? Thank you.
another possible answer There are two separate elements of a model structure - one is the "error structure" - which relates to the distribution of the error terms in your model. If you have a Gamma error structure, then your errors (or residuals) are gamma distributed, and so by extension are your responses. The other element is your linear predictor (the vector of beta parameters times your predictor or X variables), which is related to the mean of your response (mu) by the link function g(mu). The link function can be the identity, log, logit, or whatever you please. What you have identified correctly as 1/theta for the gamma distribution, is the mu (expressed here as a function of theta). This result has nothing to do with link functions - it comes from that key result about the exponential family of distributions, which says you can express any member of the exponential family in a certain form. I hope this helps.
Thanks. Exam_Machine. That helps. In fact this makes sense for me in several areas in this chapter now. Thanks again.
According to the Actuarial Tables, the canonical link function for the gamma distribution is g(mu)=1/mu. However, we do not always use the canonical link function. The log link function is much more natural in the context of premium rating because it results in a multiplicative structure which is what you would use in practice. It would also be difficult to explain to management the use of an inverse link function. I hope this helps. Duncan
