View Full Version : Part 5: Finger

Process Variance and VHM??

Oh no, not again :swear:

Truthfully, it's interesting to see how the pieces fit together, the problem is being tested on it all. :(

Wigmeister General

12-02-2003, 04:09 PM

Process Variance and VHM??

Oh no, not again :swear:

Truthfully, it's interesting to see how the pieces fit together, the problem is being tested on it all. :(

Finger is STILL on the syllabus?

He's been retired for ages.

Finger, R.J., "Risk Classification," Foundations of Casualty Actuarial Science (Fourth Edition), Casualty Actuarial Society, 2001, Chapter 6, pp. 287-342.

I have the book, but it's a pdf on the web notes too.

Was it on the syllabus in the 1800's while you were taking exams, Pops?

:P

On the topic of Finger, p 313.

I undersatnd this to mean that λ is the avg frequency of the population, and E(χ) is the avg deviation of the frequency around λ.

If that's o, should the / in the formula in the middle of the page actually be a |, meaning that it's not a division, but a conditional expected value?

If so, it's not the first misprint I've found in the Foundations :(

If not, why are we dividing by χ? :-?

The Sad Man

02-10-2006, 12:43 AM

On the topic of Finger, p 313.

I undersatnd this to mean that λ is the avg frequency of the population, and E(χ) is the avg deviation of the frequency around λ.

If that's o, should the / in the formula in the middle of the page actually be a |, meaning that it's not a division, but a conditional expected value?

If so, it's not the first misprint I've found in the Foundations :(

If not, why are we dividing by χ? :-?

I've also found his notation annoying. I think it was fairly obvious after a bit of thought that / had to equal | . I had never seen the subscript either before. Usually you'd see | χ = χ

Harbinger

02-10-2006, 09:57 AM

Wow, talk about dusting off an old thread, but you get props for taking the time to search for it.

Purple Princess

02-12-2006, 04:48 PM

I've also found his notation annoying. I think it was fairly obvious after a bit of thought that / had to equal | . I had never seen the subscript either before. Usually you'd see | χ = χ

Sad Man, did you see the errata for Finger? It corrects all that stuff.

The Sad Man

02-13-2006, 01:36 PM

Thanks, PP. I didn't think to look there but it certainly addresses the issue.

frank_exams

02-24-2006, 03:12 AM

The errata only corrects the typos and notation (which were, humorously enough, reproduced exactly in the All10 manual).

What is sorely missing is clarity on that page. I'll admit that I'm almost completely lost with his example and will probably give up by the weekend on it.

In the first paragraph on the section, Finger calls \beta the process variance, and then later in the last paragraph \lambda^2\beta is the process variance. Which is it!!? (There's only one process going on here and I think the latter is correct.)

How do we "note" that the resulting claim count is negative binomial? The prior is an arbitrary distribution with only mean and variance described!

Argh. My first upper exam and I'm already frustrated at the authors. (Not a fan of Feldblum, either.)

SouthOfSanity

02-24-2006, 10:58 AM

Argh. My first upper exam and I'm already frustrated at the authors. (Not a fan of Feldblum, either.)

Nahh really? Feldblum's "Workers' Comp Ratemaking" is a nice read..

The Sad Man

02-24-2006, 01:40 PM

Nahh really? Feldblum's "Workers' Comp Ratemaking" is a nice read..

You being sarcastic? It might not be tough reading but I can't seem to get more than 20 pages or so before the sandman pays a visit.

SouthOfSanity

02-24-2006, 03:45 PM

You being sarcastic? It might not be tough reading but I can't seem to get more than 20 pages or so before the sandman pays a visit.

I was just messing around. Though that paper really was an interesting read compared to his other papers. But I still had to break it up into parts.

I "can't wait" for the GLM paper that he co-authored that is going to be a reading for Exam 9. It is written by about 5 authors, I think, so it's huge.

frank_exams

02-24-2006, 05:12 PM

AARGH! :crying: Does anyone know how to contact Finger? Or maybe there's a FCAS strolling the boards who can help out?

I'm getting really frustrated now with p.313. I've asked a pretty sharp FCAS colleague and he wasn't able to figure out what's going on. (Initially, he wrote it off as some mixing problem from exam 4, but it was not so.)

Finger's negative binomial comment is killing me.

He claims that N has a neg. bin. distribution with paramaters \lambda and \beta, yet E(N) = \lambda and V(N) = \lambda^2\beta + \lambda?! If you use the moments to compute neg bin parameters, the outlook becomes more bleak (the parameters become {\beta}^{-1} and \beta\lambda). I convinced myself now that if I could just get P(N = 1) to look even vaguely negative binomial, then I can sleep. But,

P(N = 1) = E*(P(N = 1 | \chi)) = E*(e^{-\lambda\chi}\lambda\chi)

(E* is expectation with resepct to \chi.) But how do we compute this without the distribution of \chi?!?

I know; I should get over it if I want to pass. If only someone could give me Finger's contact info...:burn:

Howard Mahler

02-25-2006, 09:50 AM

Beginning of page 313.

chi is the expected frequency for an insured relative to average.

So for a given insured, chi times lambda is that insureds mean frequency.

Chi varies across the portfolio with mean of one and variance of beta.

Therefore lambda is the mean frequency for the whole portfolio.

(Some might prefer to rewrite mu everywhere for lambda.)

Then he assumes each insureds frequency is Poisson.

Thus each insured has a frequency process with mean and variance equal to chi times lambda.

EPV = E[chi times lambda] = lambda E[chi] = lambda.

VHM = Var[chi times lambda] = lambda^2 Var[chi]

= lambda^2 beta.

(Remember that lambda is a constant, the mean frequency for the whole portfolio.)

The final sentence on the page is wrong. The mixed distribution is not necessarily negative binomial.

I hope this helps,

Howard Mahler

frank_exams

02-25-2006, 02:45 PM

The final sentence on the page is wrong. The mixed distribution is not necessarily negative binomial.

Thanks!!!! This helps immensely. I got the right moments; I was just pointing out that a negative binomial with parameters lambda and beta would necessarily have different moments than the correct ones. Glad there are people like you strolling the boards.

Frank

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