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In the (re)insurance field, it is quite common to simulate losses via a Monte Carlo approach and a stochastic generator of both frequency and severity individually (independency between both being often assumed).

However, empirical data (on which parameters are infered) is almost often truncated (due to the deductible or retention).

Nevertheless, it seems that technicians/actuaries use - apart from the Pareto type I distribution which is quite convenient but often provides poor gof since not very flexible and/or too heavy tailed - non-left-truncated distributions such as log-normal, gamma, etc. (for severity, and Poisson or NB for claims number).

Even if the gof is amazing, is it not theoritically "wrong" to use non-left-truncated distributions for this kind of things?

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  • $\begingroup$ There are so many negatives in the question (not, wrong, not, only - each negating the other) that I don't fully understand what is actually asked :-) $\endgroup$
    – Alex
    Commented Sep 26, 2022 at 22:15
  • $\begingroup$ @Alex not sure to see why - I edited the text, removed one "not", and removed 2nd question. Hope this is clearer. $\endgroup$
    – yeahman269
    Commented Sep 27, 2022 at 8:32
  • $\begingroup$ thanks. Now it's clear. $\endgroup$
    – Alex
    Commented Sep 27, 2022 at 8:50

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I assume that retention left-truncates the severity distribution and deductible does so for both the frequency and severity distributions (even in a less predictable way that the former). If an insurance company doesn't keep (and then provides to its quants) its historical data in a form that retains the non-covered claims, or buys such truncated data, it will be in a weaker position to predict its exposure to the risk than its competitors, will more severely misprice its insurance products and be less profitable on average.

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  • $\begingroup$ Thanks for the participation. Indeed, (re)insurance companies will not be aware of the part of the data that is below retention/deductible. If one would want to model properly this kind of things, what other solution than using left-truncated distributions is available? Would a "rejection" approach relevant? (such as if drawn value is below deductible, reject and draw another one) $\endgroup$
    – yeahman269
    Commented Sep 27, 2022 at 9:09
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Even if the gof is amazing, is it not theoritically "wrong" to use non-left-truncated distributions for this kind of things?

If it is the case that some simple parametric distribution fits better than your theoretic distribution then it means that your theory is not very accurate.

It is not (entirely) wrong to go with the non-theoretic curve that performs well.

However, it is even better to adapt your theory and use a similarly more flexible curve but based on theory.

I you have a good fit with a gamma distribution or log-normal distribution then this is because of some flexibility in mixing a certain behavior in the tails with the shape of the distribution in the center that your theoretical distribution does not have. So figure out in what way your theory is too rigid and adapt it. Pareto distributions have many variants that might be suitable.

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    $\begingroup$ thanks for your comment. The way I understand your first sentence lead me to disagree with it: it is a bit like a p-value from my side: it is not because you do not reject a first model that is OK, that you actually can't find another one better. Otherwise I do align with the rest. When you say "Pareto variants", are you refering to GPD (Generalized Pareto distib)? I am also wondering if a "rejection method" would be appropriate: you redraw simulated value if it does not reach the minimum threshold (like deductible) - drawback is that it would change in the end the distrib $\endgroup$
    – yeahman269
    Commented Sep 28, 2022 at 8:48
  • $\begingroup$ I don't follow this: "it is a bit like a p-value from my side: it is not because you do not reject a first model that is OK, that you actually can't find another one better." $\endgroup$
    – Sextus Empiricus
    Commented Sep 28, 2022 at 10:28
  • $\begingroup$ I understand what @yeahman269 says here. For example, you feed a time series (which follows ARIMA(1,1,0)) into linear regression with AR(1) as a regressor (pure, without differentiation) and get a perfect R2 and p-value for the lag term. Which only proves that the relationship is spurious and doesn't make this model any better than conducted on log-differentiated time series (with times lower R2 and much worse p-value for the lag-term). $\endgroup$
    – Alex
    Commented Sep 28, 2022 at 14:59

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