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 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 not use only left-truncated distributions for this kind of context? Or a "rejection" like algorithm? (if drawn value is below threshold, then reject and draw a new one)

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?