TIA Sample Exam 4 #13 SE4 PE4

[MrActuary]

For those who do not have this question, its a K-S with left-truncated and right-censored data. Given 6 loss amounts which are fitted to an exponential and then asked to use K-S test to test the null hypothesis that the GROUND UP distribution is exponential.

The question is this... when finding the CDF (big F), the CDF of the exponential is NOT used (1 - e^-(x/theta)), but rather a modified version to account for the left-truncation.

But the question is asking about the ground up distribution. So, why do we modify the CDF?

Thanks.

[Hammer60]

The data that is given is truncated data, so we need to compare to F*.

[levenzaha]

the distribution is ground up, but you still don't know anything about what happens below the deductible. you cannot say that probability of being below a point (or range) is just the exponential because the only data you have is conditioned about being a certain point

[MrActuary]

Thanks for the replies.

The question specifically asks about the ground up distribution as being the null hypothesis. How can we test the ground up distribution if all we know about are data points above the truncation point?

The whole point of the K-S stat is to determine the fit of a distribution to the empirical data. In this case the data we have is truncated and we are asked to fit a distribution to the "ground up" data.

I feel like this may be a flawed question.

[levenzaha]

oo you didn't give the question!

The exponential is fit with an MLE. This is your best guess at the distribution, and you are then comparing it to the data t othe K-S statistic. Definitely nothing wrong with that.

although even if that wasn't the case, I still don't see a problem with fitting a parametric distribution to incomplete data. You have to do it in real life all the time! Fitting is done with whatever data you have and you do the best you can.

[Hammer60]

By adjusting the distribution F, you are comparing apples to apples and the critical values will be lowered because you have incomplete data.

[MrActuary]

Okay, I think it's becoming more clear. Appreciate the responses.

This is one of those problems where I can remember the method, but I want to understand what is going on (which is what they undoubtedly try to test us on with every problem).

The "apples to apples" comment makes a lot of sense. So even though they tell us to fit the ground up distribution, we obviously don't have ground up data since it is truncated and censored. So are we just assuming the ground up (i.e. untruncated, uncensored) data has the same distribution as the empricial data.

One last question. What if they asked us to fit a distribution to the payments, not ground up losses? I assume in this case we would not adjust for the deductible since payments are not truncated.

[Hammer60]

Originally Posted by MrActuary

So are we just assuming the ground up (i.e. untruncated, uncensored) data has the same distribution as the empricial data.

One last question. What if they asked us to fit a distribution to the payments, not ground up losses? I assume in this case we would not adjust for the deductible since payments are not truncated.

If we had an infinite amount of data that was truncated and censored we would know the exact parameters (assuming it does follow the distribution) even though we don't know about the losses below the deductible and the exact amount of the losses above the censoring. The censored and truncated data do match F* which is why we make the adjustments.

I think that is how you would approach the problem if they said that we were testing to see if the payments followed an exponential distribution, but I haven't seen any problems like that.

[levenzaha]

if you have a policy limit it would be difficult to follow any kind of single distribution but a deductible in this case would just cause u to have a shifted exponential