Approximations based on Gegenbauer orthogonal polynomials

초록

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.

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