I was looking a little bit into insurance mathematics, which left me puzzling about a question. I'll give you some context first:
When estimating the total claim amount for, say, a motor insurance, it is very common to use GLMs to model the expected costs. A standard way to go is the following:
- Build a GLM with a Poisson distribution and a log link function to model the claim frequency (i.e. estimate how many claims you'll see within 1 year)
- Build a GLM with a Gamma distribution and also a log link function to model the claim severity (i.e. estimate what is the average claim amount)
Because a log link allows you to express the desired quantity as a product of the form $$ E[Y] = exp(\beta_1 x_1) \cdot exp(\beta_2 x_2) \dots, $$ you can identify a multiplicative factor $exp(\beta_i)$ associated with each coefficient. (Here the $\beta_i$ are the coefficients of the GLM and $(x_1,\dots,x_n)$ is the, for simplicity binary, vector relating to an observation point.)
Now it is convenient and therefore customary to summarize these $exp(\beta_i)$ in so called "rating tables", as, given an observation, they allow for a quick calculation of the expected frequency (or severity) that is furthermore easy to interprete.
Assuming independence between frequency and severity, you get the total amount by multiplying the two. Now since we have log link functions in both cases, we can easily get a rating table for the total amount by simply multiplying the entries in both tables (assuming both GLMs use the same variables of course). So here is where the questions begins:
Assuming I have $m$ different pairs of frequency and severity models and I want the sum of these, can I get a rating table for this? Or mathematically, assume random variables $F_i$, $S_i$ with $i=1,...,m$ that are modelled by GLMs as above, can I derive from those coefficients a multiplicative structure for $E[B]$ with $B = F_1\cdot S_1 + F_2\cdot S_2 + \dots + F_m\cdot S_m$?
I know I could just fit a model on this random variable $B$ and use a log link function, but can I also derive such a structure from the single models?
Of course, you don't have to stick to this insurance setting, a general answer will make me just as happy!