Sampling Student's T distribution – use of the inverse cumulative distribution function

William T. Shaw

Abstract

ABSTRACT

With the current interest in copula methods, and fat-tailed or other non-normal distributions, it is appropriate to investigate technologies for managing marginal distributions of interest. We explore “Student’s” T distribution, survey its simulation, and present some new techniques for simulation. In particular, for a given real (not necessarily integer) value n of the number of degrees of freedom, we give a pair of power series approximations for the inverse, F−1 n , of the cumulative distribution function (CDF), Fn.We also give some simple and very fast exact and iterative techniques for defining this function when n is an even integer, based on the observation that for such cases the calculation of F−1 n amounts to the solution of a reduced-form polynomial equation of degree n − 1. We also explain the use of Cornish–Fisher expansions to define the inverse CDF as the composition of the inverse CDF for the normal case with a simple polynomial map. The methods presented are well adapted for use with copula and quasi-Monte-Carlo techniques.