Actuarial Outpost Buhlmann cred for r.v.'s X1, X2, X3
 Register Blogs Wiki FAQ Calendar Search Today's Posts Mark Forums Read
 FlashChat Actuarial Discussion Preliminary Exams CAS/SOA Exams Cyberchat Around the World Suggestions

 United KingdomActuarial Jobs General Insurance, Life & Other Areas Canada Actuarial JobsCasualty, Life,Pension,Health & Investment Entry LevelActuarial JobsAll Disciplines Health, Life, Pension, Casualty, Investment D.W. Simpson & Company ActuarialRecruitersWorldwide www.dwsimpson.com

#21
05-18-2006, 11:35 AM
 NewTubaBoy Member CAS Join Date: Jul 2004 Location: Boston Studying for CAS 8 Favorite beer: Blue Moon Posts: 1,391

I pray for no thrown out questions.... because it just ups the stupid pass score.
__________________
He slept a summer by my side
He filled my days with endless wonder
#22
05-18-2006, 12:20 PM
 Abraham Weishaus Member SOA AAA Join Date: Oct 2001 Posts: 6,200

Quote:
 Originally Posted by Tom Craven The problem didn't state that X1, X2, X3 have the same mean, and, they were random variables, not observations of a single random variable. I think they weren't playing the role of past losses from policyholders of unknown classes, which are labeled X1, ... , Xn in the statement of the Buhlmann model on p. 557 of Loss Models. That would have been a list of past losses like X1=2, X2=6, X3=8, where X1, ... , Xn are i.i.d conditional on the class. As far as I could tell, in the test problem X1, X2, X3 are just 3 random variables, unrelated except for happening to have the same variance. For example, X1 could be the r.v. representing a loss from a policyholder of class 1 (similarly for X2, X3). If we had see 3 years of loss data for a policyholder of class 1, that would have been 3 observations, all from the distribution of X1. (We could then have applied the Buhlmann model, with mu = E(E(Xi|i)) and n = 3, to determine a credibility estimate for the next observation of X1.) But when they state, "Given an observation of X1 of (some number), the Buhlmann cred estimate of X2 is 1.4," it seems to me that the observation of (some number) of X1 couldn't have been used in the usual Buhlmann formula with n=1 to produce the credibility estimate for the next obs of X2, since X1 and X2 are not (necessarily) identically distributed. My interpretation was that the estimate 1.4 was produced from the Buhlmann formula with n=0, for the 0 observations of X2 that we are given. (I mean that in that sentence we are not yet given an observation of X2; later in the problem an observation of X2 is given.). So with n=0, Z=0, and 1.4 is mu = E(E(Xi|i)).
Unfortunately I'll never see the exact wording of the problem, but I doubt this interpretation is correct. One usually wouldn't refer to a Buhlmann credibility estimate when there is by design no credibility. (It could happen "accidentally" if v is infinite or a is 0.)

From what I read here, it sounds like 2 was the total variance (not the EPV), and they were testing you on backing out v and Z.
#23
05-18-2006, 12:21 PM
 Antimony Member CAS Join Date: Nov 2001 Location: St. Louis, MO Posts: 114

Quote:
 Originally Posted by Tom Craven I interpret "Var(Xi) = 2 for all i" as meaning the same thing as "the process variance is 2 for each of the 3 classes."
That's how I read it too. I don't really see how 2 could be the TV, because the wording of the phrase made it clear that it was talking about 3 different variances that were all equal to 2.
#24
05-18-2006, 12:24 PM
 TwinsFan1987 Member SOA Join Date: Jun 2003 Location: Chicago Studying for FSModules/waiting for APM College: Northwestern College (IA) alumnus Posts: 676

__________________
Quote:
#25
05-18-2006, 01:14 PM
 teapeaexcueexplustea Member SOA Join Date: Mar 2006 Location: Ontario, Canada Studying for DMAC & CSP-IC Posts: 522

When I said above that I had done the same thing, I actually misread the post. I saw that Var(Xi) was the process variance for each X, therefore the EPV=Var(Xi)
#26
05-18-2006, 01:16 PM
 Livan33 Member SOA Join Date: May 2005 Favorite beer: Lucky Lager Posts: 1,652

Quote:
 Originally Posted by TwinsFan1987 Do both interpretations lead to a given answer?
Were there actual answers for this question? Or was it just ranges? I can't seem to remember.

Never mind, I just saw that Tom said his answer fell into one of the ranges.

Last edited by Livan33; 05-18-2006 at 01:18 PM..
#27
05-18-2006, 01:18 PM
 pi_rsquare Member SOA Join Date: Nov 2005 Location: Singapore Posts: 45

I used Var(Xi) as Total Variance and I got 1.44 as the answer.

If you did it with other method, what answer did you get ?
__________________

Never be bullied into silence.
Never allow yourself to be made a victim.
Accept no one's definition of your life.
Define yourself.

#28
05-19-2006, 01:36 PM
 Surfohio Member Join Date: Apr 2005 Posts: 1,645

Here is something interesting I tried.

The estimate of x3, assuming it has some credibility from past observations is

x3 = 2/(2+k)*1.25 + k/(2+k) * mu
where 1.25 is the average of actual observations x1 and x2.

Then, since 1/(1+k)*1 + k/(1+k)*mu = 1.4 <-- estimate of x2

mu = (1.4 - 1/(k+1))/(k/(k+1)) = (1.4k + .4)/k

Therefore

x3= 2.5/(2+k) + (1.4k+.4)/2+k = (2.9 + 1.4k)/(2+k) which is always between 1.4 and 1.45 regardless of k (since k is always positive).

Anyone remember the choices for this one?
#29
05-19-2006, 04:17 PM
 Tom Craven Join Date: May 2006 Posts: 10

Quote:
 Originally Posted by Surfohio Anyone remember the choices for this one?
I THINK that A was: lower than 1.5. Then the remaining choices were increments of .1, up to E being everything above, of course.

 Thread Tools Display Modes Linear Mode

 Posting Rules You may not post new threads You may not post replies You may not post attachments You may not edit your posts BB code is On Smilies are On [IMG] code is On HTML code is Off

All times are GMT -4. The time now is 06:30 PM.

 -- Default Style - Fluid Width ---- Default Style - Fixed Width ---- Old Default Style ---- Easy on the eyes ---- Smooth Darkness ---- Chestnut ---- Apple-ish Style ---- If Apples were blue ---- If Apples were green ---- If Apples were purple ---- Halloween 2007 ---- B&W ---- Halloween ---- AO Christmas Theme ---- Turkey Day Theme ---- AO 2007 beta ---- 4th Of July Contact Us - Actuarial Outpost - Archive - Privacy Statement - Top