Actuarial Outpost "Adjusting" A compound Negative Binomial Distribution
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#1
12-04-2005, 05:10 PM
 MyKenk Note Contributor Join Date: Nov 2005 Location: http://mhenk.blogspot.com Studying for CAS 9/8/whatever? College: Drake '06 Posts: 8,519
"Adjusting" A compound Negative Binomial Distribution

Hey guys. I have the following as a homework problem, and I know the general idea of what I need to do, but I'm having some trouble, so any help would be greatly appreciated:

After imposing a deductible, claim amounts are:
0 with prob .6
1 with prob .2
2 with prob .2

PRIOR to imposing the deductive, (N) number of losses are from a negative binomial distribution, r=2, beta=2.

I need to find aggregate probabilites for S...

I have already done this when N (number of claims) was a poission distribution, and that involved adjusting lambda and the claim amount probabilities, so I'm assuming there's something similar I can do with the negative binomial, but I can't seem to figure out how to adjust the number of claims distribution.

I would like to be able to use the recursive formula derived in section 12.4 of bowers, that's what I did with the poisson, but I know that that recursion does not work when the claim amounts distribution places probability at zero, so naturally, an adjustment came to mine.

Again, any help would be terrific.
#2
12-04-2005, 10:46 PM
 sam_broverman Member Join Date: Feb 2005 Posts: 638

The adjusted claims distribution will be negative
binomial with the same r = 2, but an adjusted beta.
The adjusted beta is the original beta multiplied
by the probability that the original loss is above the deductible
(that prob. is .4 in this case), so the adjusted beta is .8 .
Then aggregate payments S (after deductible) has
a compound distribution with neg binomial frequency
with r = 2 and beta = .8 and a severity distribution which
is the conditional distribution of X-d given X>d .
From the way you have described things, it
looks like this conditional dist is a two-point distwith outcomes
1 (prob. .5) and 2 (prob. .5) .
#3
12-04-2005, 11:30 PM
 MyKenk Note Contributor Join Date: Nov 2005 Location: http://mhenk.blogspot.com Studying for CAS 9/8/whatever? College: Drake '06 Posts: 8,519

awesome... thanks a lot.

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