CT 6 : Q & A Banks Part 1 - Q 1.31 (ii) (a) Formula for estimation on page 24 / chapter 4 is: L(theta) = ∏ (from 1 to n) f (x,theta) X [1-F(M;theta)]^m where n = no. of claims known exactly = insurer's claim m = no. of claims above retention value = reinsurer's claim BUT in the solution formula used is : L = [F(M)]^n X ∏ (from 1 to m) f(x) WHY ???
This is MLE with censored data F(x,theta) = F(x) this is just notation Then, whether we use F(M) or 1-F(M) depends on what we know about the claim, ie how it is censored. If we know the claim is less than M we use F(M). If we know the claim is greater than M we use 1-F(M). The n in the formula you quoted is the number of items of censored data. f(x) is the "probability" that the claim size = x and this is what we usually use in our likelihood. So, if we don't know the claim size then all we can use is the probability of what we do know, ie F(M) or (1-F(M)). Try to think of it like this rather than learning formulae. Good luck! John
