This afternoon, an interesting point was raised, and I wanted to get back on it (since I did publish a post on that same topic a long time ago). How can we adapt a logistic regression when all the observations do not have the same exposure. Here the model is the following: ,
- the occurence of an event on the period is unobserved
- the occurence of an event on is observed (as well as )
If we assume that the ‘occurence of an event’ is the first occurence of a Poisson processus, we can prove that
i.e. no event occur on if no event occur on and no event occur on . Assuming independence between the two, we can prove that we have
With words, it means that the probability of not having a claim in the first six months of the year is the square root of not have a claim over a year. Which makes sense.