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Transformed Jacobi polynomial density and distribution approximations
Journal of the Korean Data & Information Science Society 2018;29:1087-93
Published online July 31, 2018
© 2018 Korean Data and Information Science Society.

Hyung-Tae Ha1

1Department of Applied Statistics, Gachon University
Correspondence to: Associate professor, Department of Applied Statistics, Gachon University, Sungnam-ci, Kyunggi-do 461-701. R. Korea. E-mail: htha@gachon.ac.kr
Received May 12, 2018; Revised June 26, 2018; Accepted July 6, 2018.
This is an Open Access article distributed under the terms of the Creative Commons Attribution Non-Commercial License (http://creativecommons.org/licenses/by-nc/3.0) which permits unrestricted non-commercial use, distribution, and reproduction in any medium, provided the original work is properly cited.
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 corresponding 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.
Keywords : Density approximation, generalized beta distribution, Jacobi polynomials, transformation.