Credibility theory notes question

[nick.campbell]

Hello, on page 17, chapter 5, CT6 notes, it reads...

"In fact what is required is E[X|x]". It then goes on to explain why another solution is not correct, rather than why this solution is correct.

Is this a mistake? I'm thinking that should read E[X|lamda]?

If it is correct, could you please show me how it relates to the credibility estimate? This is done quite eligantly in the normal/normal model, which makes me think the poisson/gamma should be similar.

Thanks,

Nick

 

[nick.campbell]

anyone?

 

[johnmunge]

Let me have a try at it;

The Poisson/Gamma model is used to model the number of claims.

What we require is the expected number of claims in the jth year.

Random variable X represents the number of claims jth year.

X/lambda ~ Poisson(lambda), whereby lambda is random with Gamma(alpha, beta) distribution.

We would naively think that the solution to our problem is E(X) = E[E(X/lambda)]= alpha/beta= constant

But what we actually need is the expectation of X given that we have past data on it: ie we require E[E(X/lambda)/x(vector)] as written on the notes, which translates to the mean of the posterior. This means that expected number of claims will be more closer to the actual figure the more observed data we have on random variable X.

I hope that helps, i might be wrong

 

[John Lee]

Hello, on page 17, chapter 5, CT6 notes, it reads...

"In fact what is required is E[X|x]". It then goes on to explain why another solution is not correct, rather than why this solution is correct.

Is this a mistake? I'm thinking that should read E[X|lamda]?

If it is correct, could you please show me how it relates to the credibility estimate? This is done quite eligantly in the normal/normal model, which makes me think the poisson/gamma should be similar.

Thanks,

Nick

Apologies Nick for the delay. I had a messy accident with a circular saw and have been out of action until now and the CT6 queries didn't get reassigned.

I think it should say we want to estimate E[E(X|λ)|x_] which is equivalent to E[λ|x_] which is the mean of the posterior which would be (Σx+α) / (n+β).

 

[nick.campbell]

Thanks for your reply john and John - my mind is racing as to the severity of a circular saw accident! Hope you're ok.

That makes more sense now, will look over my notes when I get home.

Cheers!

 

[John Lee]

Thanks for your reply john and John - my mind is racing as to the severity of a circular saw accident! Hope you're ok.

That makes more sense now, will look over my notes when I get home.

Cheers!

Glad it helped.

You're welcome to look at the post-op pictures on my twitter account @actuarial_tutor (pre op rather too messy for public display...)