
#1




SOA #29
Hi, I tried doing this question a different way. Namely, using the Buhlmann credibility estimate formula: Pc=Zxbar+(1Z)uhat. In this situation, is using the Buhlmann credibility estimate viable? I got: xbar=3/6 v=EPV=E[Var(Xtype of risk)]=0.119 a=VHM=Var(E[Xtype of risk])=0.0085. uhat=E[E[Xtype of riks]]=0.15. Thank you for your help. 
#2




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.

#4




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




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




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