Question is set up as follows:
Claim counts follow poisson with mean theta.
Given theta, x has an inverse exponential distribution with mean 10theta.
Theta varies by policy holder according to a single parameter pareto, alpha =5, and other parameter equal to 1.
We have 5 years of experience for one policyholder.
Table gives claim counts for each year and the claim sizes in each year.
With 3 counts, 1 in year 4 and 2 in year 5.
Calculate the probability that the same policyholder's next claim is of size 10 or greater.
The solution proceeded to find the likelihood used in the posterior as a product of both the given claim sizes and the claim counts. My question is why do we need to account for claim counts as well if we're only interested in individual claim sizes? Is it because both the count and sizes distribution depend on the theta, so we need to incorporate all the information we have on theta. My other guess was because the question is inherently asking for the probability of the aggregate claim size to be 10 or greater and the trick was here that in the aggregate there is only 1 person but with multiple years of data but seems unlikely this is the case. I'm unsure how if we want the probability of the same policyholder's next claim to be of size 10 or greater why we need to involve claim counts to find our posterior of claim sizes.
Claim counts follow poisson with mean theta.
Given theta, x has an inverse exponential distribution with mean 10theta.
Theta varies by policy holder according to a single parameter pareto, alpha =5, and other parameter equal to 1.
We have 5 years of experience for one policyholder.
Table gives claim counts for each year and the claim sizes in each year.
With 3 counts, 1 in year 4 and 2 in year 5.
Calculate the probability that the same policyholder's next claim is of size 10 or greater.
The solution proceeded to find the likelihood used in the posterior as a product of both the given claim sizes and the claim counts. My question is why do we need to account for claim counts as well if we're only interested in individual claim sizes? Is it because both the count and sizes distribution depend on the theta, so we need to incorporate all the information we have on theta. My other guess was because the question is inherently asking for the probability of the aggregate claim size to be 10 or greater and the trick was here that in the aggregate there is only 1 person but with multiple years of data but seems unlikely this is the case. I'm unsure how if we want the probability of the same policyholder's next claim to be of size 10 or greater why we need to involve claim counts to find our posterior of claim sizes.
Posterior of claim sizes, need to account for claim counts in posterior?
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