Transformed Jacobi polynomial density and distribution approximations

Abstract

A density approximation method by making use of transformed Jacobi orthogonal polynomials is developed for approximating the density and distribution functions of random variables with various compact supports. The method is formulated by a product of a generalized Beta density function and a linear combination of the cor-responding Jacobi orthogonal polynomials, where the infinite sequence of the Jacobi orthogonal polynomials can be generated from the initial approximation of the generalized Beta density function. The moment matching technique was used to estimate parameters of the initial approximation and the coefficients of the linear combination in terms of the exact moments of the target distribution. The numerical examples using an artificial mixture of non-standard density functions and a test statistic show that the proposed method provides excellent density and distribution approximants