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This paper is devoted to a fractional generalization of the Dirichlet distribution. The form of the multivariate distribution is derived assuming that the npartitions of the interval [0, Wn] are independent and identically distributed random variables following the generalized Mittag-Leffler distribution. The expected value and variance of the one-dimensional marginal are derived as well as the form of its probability density function. A related generalized Dirichlet distribution is studied that provides a reasonable approximation for some values of the parameters. The relation between this distribution and other generalizations of the Dirichlet distribution is discussed. Monte Carlo simulations of the one-dimensional marginals for both distributions are presented.

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 a normalization condition is preserved. 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 κ -statistics might be 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. • An approximation of the first passage time probability density function is proposed • The proposed approximation is of Laguerre-Gamma polynomial type • The resulting recursive nested algorithm involves FPT moments. • A suitable normalization condition is used as stopping criterion. • Estimation of the parameters is proposed through a likelihood function. • Approximations are computable even if only FPT sample data are available. • The special case of Geometric Brownian Motion is analyzed. [ABSTRACT FROM AUTHOR]