  This paper analyzes a method to approximate the first passage time
probability density function which turns to be particularly useful if only
sample data are available. The method relies on a Laguerre-Gamma polynomial
approximation and iteratively looks for the best degree of the polynomial such
that the fitting function is a probability density function. The proposed
iterative algorithm relies on simple and new recursion formulae involving first
passage time moments. These moments can be computed recursively from cumulants,
if they are known. In such a case, the approximated density can be used also
for the maximum likelihood estimates of the parameters of the underlying
stochastic process. If cumulants are not known, suitable unbiased estimators
relying on k-statistics are employed. To check the feasibility of the method
both in fitting the density and in estimating the parameters, the first passage
time problem of a geometric Brownian motion is considered.
