I think I have an answer for your first question.
The reason the lilkelihood function isn't what you thought,
(Product from 1 to 50 of [ f(yi;theta) / F(k;theta) ] )
is that even though the data is truncated, the losses at or above k are
still recorded. If you look at the discussion on page 164 of Hogg and
Klugman regarding truncation from above, they are referring to a situation
where losses above the limit are not recorded.
In this case, the modifed p.d.f. (I'll denote it as g(x;theta)) is:
g(x;theta) = f(x;theta) for x < k
Pr(x=k) = 1 - F(k;theta)
The likelihood function then consists of two pieces, the part for losses yi
< k (which contribute f(yi;theta) each), and the part for losses yi >= k
(which contribute (1 - F(K;theta) each). Putting it all together gives the
quoted result.
L(theta)=Product from 1 to 50 of f(yi;theta) times [1-F(k;theta)]^75
Dave Kennerud
kennerud@concentric.net
----------
> From: Dimitra Roidakis <dr@g-g-a.com>
> To: studygroup4b@lists.casact.org
> Subject: Study Manual 4B, Fall 1998
> Date: Monday, September 14, 1998 8:45 AM
>
> I have 2 questions from the Study Manual 4B, Fall 1998
> Unfortunately, I don't know the corresponding question number from
> the SOA manual.
>
> First, concerning truncation, Q.B20 page 367
>
> For type Y policies they are truncated from above and there are 50
> losses less than k and a total of 75 losses that exceed k and have
> been recorded. they ask you for the ml estimate of theta.
> the answer they give you is as follows:
>
> L(theta)=Product from 1 to 50 of f(yi;theta) times [1-F(k;theta)]^75
> Why isn't Product from1 to 50 of[ f(yi;theta) / F(k;theta) ]???
>
>
> Dimitra
> dr@g-g-a.com