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?

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.
