Actuarial Outpost SOA #29
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 Short-Term Actuarial Math Old Exam C Forum

#1
06-19-2018, 02:33 PM
 cashcrazy CAS Join Date: Nov 2016 College: Waterloo Posts: 11
SOA #29

Hi, I tried doing this question a different way. Namely, using the Buhlmann credibility estimate formula: Pc=Zxbar+(1-Z)uhat. In this situation, is using the Buhlmann credibility estimate viable?

I got:
xbar=3/6
v=EPV=E[Var(X|type of risk)]=0.119
a=VHM=Var(E[X|type of risk])=0.0085.
uhat=E[E[X|type of riks]]=0.15.

#2
06-19-2018, 02:39 PM
 Academic Actuary Member Join Date: Sep 2009 Posts: 8,386

The question asks for the posterior probability. If it had asked for the Buhlman credibility estimate you would have used that. The two estimates are not the same except for special cases.
#3
06-19-2018, 02:48 PM
 cashcrazy CAS Join Date: Nov 2016 College: Waterloo Posts: 11

Would you be able to tell me about the special cases? Thank you.
#4
06-19-2018, 02:48 PM
 jas1290 Member CAS Join Date: Mar 2014 Studying for MAS-I Favorite beer: Yuengling Posts: 143

The Buhlmann credibility estimate would not be equal to the posterior probability here. A more detailed explanation is that the EPV and VHM are calculated prior to knowing the particular observation(s), so since we want the poster probability; this leads to the wrong answer if calculated using Buhlmann credibility estimate.

Buhlmann = Bayesian for Poisson/Gamma, Binomial/Beta, Normal/Normal. Basically in a conjugate prior situation.
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#5
06-19-2018, 04:26 PM
 Academic Actuary Member Join Date: Sep 2009 Posts: 8,386

In general the likelihood has to be linear exponential. The other possibility would be exponential/gamma or exponential/inverse gamma depending upon whether the exponential parameter was in the numerator or denominator.
#6
06-19-2018, 05:19 PM
 Jim Daniel Member SOA Join Date: Jan 2002 Location: Davis, CA College: Wabash College B.A. 1962, Stanford Ph.D. 1965 Posts: 2,698

Quote:
 Originally Posted by jas1290 The Buhlmann credibility estimate would not be equal to the posterior probability here. A more detailed explanation is that the EPV and VHM are calculated prior to knowing the particular observation(s), so since we want the poster probability; this leads to the wrong answer if calculated using Buhlmann credibility estimate. Buhlmann = Bayesian for Poisson/Gamma, Binomial/Beta, Normal/Normal. Basically in a conjugate prior situation.
Careful. Not in ALL conjugate priors.
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