Tag Archives: Shaffer

Modelling Occurence of Events, with some Exposure

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 https://latex.codecogs.com/gif.latex?Y_i^\star on the period https://latex.codecogs.com/gif.latex?[0,1] is unobserved
  • the occurence of an event https://latex.codecogs.com/gif.latex?Y_i on https://latex.codecogs.com/gif.latex?[0,E_i] is observed (as well as https://latex.codecogs.com/gif.latex?E_i)

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.

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