Modelling losses in insurance - why nobody seems to talk about left-truncated distribution? 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?
 A: 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.
A: 
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.
