I've had a search for the answer to this problem but I can't find a solution anywhere.
I am using a Poisson model and estimating the average time of a potential insurance claim arising in the future. There are no claims occurring at the moment and I have a mean time of 10 years from now, based on lots of external factors (such as media coverage, legal rulings for possible claimants etc).
In one years' time will the mean time to claim be reduced to 9 years, considering all of the external factors have stayed the same and there's still no claims being reported?
I can see why it would. However, in ten years time this would mean the mean time is now 0 even if there are still no claims arising even though the risk of some claims arising should all external factors be the same will still be greater than zero.
Therefore I would like to know how the mean of the Poisson distribution should reduce every year in these circumstances.
If anybody could assist it would be much appreciated. Thanks
If the number of claims in fixed time has Poisson distribution, then the wait time to a claim has exponential distribution.
By the memorylessness of the exponential, the mean time to claim, given there has been no claim for a year, remains at $10$ years.