Gross/net claim amount in MLE

[Bharti Singla]

Hii all
I'm sharing two qus. here, one if of Q&A part1 , qus.1.25 and second it IAI, May 2015 qus 5. My query is:

Qus1.25 - Here we have XOL reinsurance arrangement and only the info of claims which are greater than retention level is given, so we use the conditional dist. of reinsurer to find the MLE of ∅. And xi is given here the gross claim amount before reinsurance. And they have used this gross amt. while calculating the MLE.

Qus.5(b), IAI- Here we have a policy excess in force and only the info of claims which are greater than the excess limit is given, so we use conditional dist. of insurer to find the MLE of ∅. And xi given here is the net claim amount after deducting the excess. And they used this net amt. while calculating the MLE.

So, what's the difference in both qus? Why using different approaches in both? The xi we use in conditional dist. pdf in the eq. of MLE, is the total/gross claim amt. or the net claim amount?

Please anyone help asap.
I know its bit time consuming to answer it. But plz could anyone try?
Thankyou.

 

[vgarg]

Using the same notations from the core readings lets start by the pdf of reinsurer's conditional claims distribution which is

\(g_{W_i}(w) = \dfrac{f_{X_i}(w+M)}{1-F_{X_i}(M)} \quad \text{for } w>0 \Leftrightarrow x-m>0\)

Here \(x_i\) is a gross claim amount while \(w_i\) is a claim amount after deducing \(M\) on \(i^{th}\) claim.

Now there can be two forms in which the data of claims that involved reinsurer is given :

• In the form of \(x_i's\) i.e. only those gross claim amounts in which reinsurer is involved or in which \(x_i's\) are all greater than \(M\). In this case the likelihood function to estimate the parameter of the distribution of the claims data would be : \begin{align*} l(\theta) &= \prod_{i=1}^{n}g_{W_i}(w_i)\\ &= \prod_{i=1}^{n}\dfrac{f_{X_i}(x_i)}{1-F_{X_i}(M)} \end{align*}
• In the form of \(w_i's\) i.e. only those claims, that involved reinsurer or that exceeded \(M\), after deducing excess \(M\). In this case the likelihood function to estimate the parameter of the distribution of the claims data would be : \begin{align*} l(\theta) &= \prod_{i=1}^{n}g_{W_i}(w_i) \end{align*}

Now try those two questions by using the notations above, I am sure your confusion would be resolved.