# Best distribution for insurance restitution

I wish to discover the best distribution to model non-zero restitution* values of insurance as a response variable in a regression.

The problems I find here:

• Knowing that the data came from insurance restitution doesn't seem bring me any information about its distribution.
• Any restitution value is valid in theory, but in fact, all data have discrete values. So should I model it as discrete or as continuous?

So far, I thought of using zero-truncated negative binomial, gamma and log-normal.

(*) Is restitution the word English speakers use for this?

• It's difficult to know for sure what word to suggest without you explaining what it is you're referring to. Are you talking about the money insurers pay to insured parties in the event of a successful claim? Feb 7, 2014 at 23:48
• If so, terms like 'claim payment' or 'paid loss' might apply, but there are a number of other terms. Feb 8, 2014 at 0:01
• I'd lean toward 'paid loss', because 'claim payment' can sometimes refer to a single installment of a series of payments insurers might make in respect of a single claim as the claim evolves. However, if you simply define what you mean by 'restitution' there's no problem with using that. Feb 8, 2014 at 0:15
• I used restitution because a direct translation of what we use in Portuguese. It seems that paid loss is more common in the literature, so I'll use with it from now on. Feb 8, 2014 at 0:19

There's no one 'best distribution' for these things; the specifics are impacted by so many different things, so there's no 'simple' form that will be quite right, it's a matter of whether you can get a satisfactory approximation (though in many cases you don't need to identify a specific distribution, it depends on what questions you're trying to answer).

There's whole books* dealing with loss distributions, so I can't hope to give a comprehensive list and discuss the various characteristics.

A few commonly used distributions in various situations include gamma, lognormal, inverse Gaussian, Pareto, log-logistic - and truncated (or sometimes, censored) versions of each of those in some cases (e.g. fixed deductibles lead to truncation). You see many other distributions as well. In practice, the payments are a mixture of different risk classes, with different loss-size characteristics, and none of those are especially suitable in many cases.

Where sums-insured tend to occur at round numbers (though the losses might not be round numbers), mixture distributions might be used.

A first thing to do might be to take logs, and see what that looks like. But if you're going to use a histogram, use more classes! If you use only a few classes for a histogram with data that tends to be a bit clumpy, you can get some very different impressions, depending on your choices of class width and class origin, in extreme cases, even to looking either left skew or right skew with only slightly different choices.

* For example:

R. V. Hogg and S. A. Klugman (1984),
Loss Distributions
Wiley, New York.
ISBN 0-4718792-9-0

S. A. Klugman, H. H. Panjer, and G. Willmot (2004),
Loss Models: From Data to Decisions
Wiley, New York, 2nd Ed.
ISBN 0-4712157-7-5

• I knew best distribution wasn't the best word for that, but didn't thought of anything else in the title. But do think a continuous distribution is a better strategy here? Feb 8, 2014 at 0:09
• All the distributions I suggested are continuous. It would be rare to model claim payments as discrete (except when the actual amounts are limited to a few possible values, or for some numerical/calculational benefit). NB I updated my answer a little Feb 8, 2014 at 0:11
• Using log things get a little more normalized, how haven't thought of this earlier =]. My mistake using the default number of classes of R too. Thanks a lot for the help. Feb 8, 2014 at 0:25
• Since you're an R-user, you might find this vignette of some value. Feb 8, 2014 at 0:28