CT6 Q&A 1.25

[Bharti Singla]

Hii all
Here, it is given that the reinsurer assumes that X has a weibull distribution where X is the gross claim amt. before reinsurer. In part (ii), (a) the MLE of ∅ is calculated in reinsurer's point of view. It is given as:
L(∅)= π f(xi,∅)× [F(M)]^600
(where i goes from 1 to 50)

Given that 600 claims are below retention and 50 claims are above retention.
And if I calculate MLE of ∅ as:
(insurer's point of view)

L(∅) = π f(xi,∅) × [P(X>M)]^50
(where i goes from 1 to 600)
I'm getting a different answer by this method.

Is the reason that the dist. of X given here is assumed by the reinsurer? The actual claim amount dist. may be different?

 

[vgarg]

Hii all
Here, it is given that the reinsurer assumes that X has a weibull distribution where X is the gross claim amt. before reinsurer. In part (ii), (a) the MLE of ∅ is calculated in reinsurer's point of view. It is given as:
L(∅)= π f(xi,∅)× [F(M)]^600
(where i goes from 1 to 50)

Given that 600 claims are below retention and 50 claims are above retention.
And if I calculate MLE of ∅ as:
(insurer's point of view)

L(∅) = π f(xi,∅) × [P(X>M)]^50
(where i goes from 1 to 600)
I'm getting a different answer by this method.

Is the reason that the dist. of X given here is assumed by the reinsurer? The actual claim amount dist. may be different?

You are not utilizing the info given in the question properly.
In the first we are only given the info of the claims that exceeds the retention limit i.e. in which reinsurer is involved we are not given any info about the claims in which reinsurer isn't involved. Therefore here to estimate the parameter we need to use the conditional claims distribution in the likelihood function.
In the second part we have additional info that completes the information about our claims data and we need to use this information as well in our likelihood. Therefore instead of using conditional distribution we use here the unconditional distribution i.e.

\( l(\theta) = \prod_{i=1}^{50} f(x_i;\theta) \times \big (\Pr(X_i < M) \big )^{600}\)

 

[Bharti Singla]

You are not utilizing the info given in the question properly.
In the first we are only given the info of the claims that exceeds the retention limit i.e. in which reinsurer is involved we are not given any info about the claims in which reinsurer isn't involved. Therefore here to estimate the parameter we need to use the conditional claims distribution in the likelihood function.
In the second part we have additional info that completes the information about our claims data and we need to use this information as well in our likelihood. Therefore instead of using conditional distribution we use here the unconditional distribution i.e.

\( l(\theta) = \prod_{i=1}^{50} f(x_i;\theta) \times \big (\Pr(X_i < M) \big )^{600}\)

Thankyou. I got this, but it's not actually my doubt. I got the first part. I am asking that in part (ii), since we now got the complete info of the claims which are below retention as well as which are above retention. Then can't we find MLE like this:

L(∅)= π f(xi;∅)× [P(X>M)]^50
(i goes to 1 to 600)
where X is the dist. of total claim amounts.

I am relating it with this given in pic below (page24 of ch4). Here, m no. of claims above retention and n no. of claims below retention. So, for this qus. m= 50 and n=600.

 

[vgarg]

No we can't do it like this because we have the data for those claims that exceeded the retention limit i.e. if \(M=900\) then we are given data of the form \(\underset{50 \text{ data points }}{\underbrace{909,987,1267,\ldots}}\) and we can utilise this info only when we use likelihood of the form

\(
l(\theta) = \underset{\text{ utilizes the given data }}{\underbrace{\prod_{i=1}^{50} f(x_i;\theta)}} \times \big (\Pr(X_i <M) \big )^{600}
\)

We can't use the expression

\(
l(\theta) = \underset{\text{We don't have data! }}{\underbrace{\prod_{i=1}^{600} f(x_i;\theta)}} \times \big (\Pr(X_i >M) \big )^{50}
\)
because it requires the data of the claims that didn't involve reinsurer..

 

[Bharti Singla]

Ohh, Okay. Now I got it.
Thankyou so much!