Approximations based on Gegenbauer orthogonal polynomials†
Hyung-Tae Ha1
1Department of Applied Statistics, Gachon University
Correspondence to: † This research was supported by the MSIT (Ministry of Science and ICT), Korea, under the ITRC (Information Technology Research Center) support program (RS-2023-00259004) supervised by the IITP (Institute for Information & Communications Technology Planning & Evaluation).
1 Professor, Department of Applied Statistics, Gachon University, Sungnam 13120, Korea. E-mail: htha@gachon.ac.kr
1 Professor, Department of Applied Statistics, Gachon University, Sungnam 13120, Korea. E-mail: htha@gachon.ac.kr
Received May 13, 2024; Revised May 31, 2024; Accepted June 11, 2024.
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
- This paper presents an advanced methodology for approximating the density functions of random variables with compact support, utilizing Gegenbauer orthogonal polynomials. The proposed approach expresses the density as a product of a weight function and a linear combination of Gegenbauer polynomials. We apply the moment matching technique to estimate these coefficients, ensuring the approximation accurately reflects the target distribution’s exact moments. The explicit expressions for the coefficients of the linear combination in the density estimator, the Gegenbauer polynomial coefficients, the normalizing constant, and the orthogonality factor are also provided in this paper. Furthermore, a transformation method is used to generalize the compact support interval [−1, 1] to [a, b], and the corresponding transformations are applied accordingly. Numerical experiments validate the stability and accuracy of the proposed method in approximating complex density functions.
- Keywords : Density approximation, Gegenbauer polynomials, moment matching, orthogonal polynomials, transformation method
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