Chebyshev approximation of the distribution function of nonnegative random variables
type: Item
instance of: scholarly article
title: Chebyshev approximation of the distribution function of nonnegative random variables
zbMATH Open document ID: 0881.41009
DOI: 10.1007/BF02366534
author: I. M. Kovalenko
author: A. N. Nakonechnyj
author: A. B. Romanov
published in: Cybernetics and Systems Analysis
publication date: 1998-02-17T00:00:00Z
review text: Let \(F(z)\), \(z\geq 0\), be a distribution function, \(F(0)=0\), \(F(+\infty)=1\) and denote \(f(x):= F(\ln(2/(1-x))\), \(x\in [-1,1]\). If \(F\) is continuous, consider the Chebyshev expansion of the function \(f\) \[ f(x)= (1/2)a_0(f)+ \sum_{k=1}^\infty a_k(f)T_k(x), \qquad\text{where}\quad x\in[-1,1], \] \(T_k\) are the Chebyshev polynomials and \[ a_k(f)= (2/\pi) \int_{-1}^1 (1-x^2)^{-1/2} f(x)T_k(x)dx, \qquad k\geq 0. \] Denote \(\varphi(j):= \int_0^\infty \exp(-jx)dF(x)\), \(j\geq 1\). By using the partial sums of the Chebyshev expansion of \(f\), one derives linear systems for computing the coefficients \(a_k(f)\) in terms of the numbers \(\varphi(j)\). Then, by retaining the partial sums of the Chebyshev series, one obtains approximations for \(F(z)\). The sharpness of this type of approximation is pointed out in the case of approximation with four terms, by considering some examples and comparing them with other methods.
Mathematics Subject Classification ID: 41A10
Mathematics Subject Classification ID: 41A60
Mathematics Subject Classification ID: 42C10
Mathematics Subject Classification ID: 62E20
zbMATH DE Number: 1051115
zbMATH Keywords: distribution function
zbMATH Keywords: Renyi theorem
zbMATH Keywords: Chebyshev expansion
reviewed by: Radu Paltanea
MaRDI profile type: Publication
cites work: Q3026020
cites work: Q4076022
cites work: Q3969669
cites work: Q4147381
cites work: Q4197802
cites work: Q3993247
cites work: Q3898366
cites work: Approximation of distribution function of additive functional of Wiener process by segment of Chebyshev series
full work available at URL: https://doi.org/10.1007/bf02366534
OpenAlex ID: W1996157668
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description: scientific article; zbMATH DE number 1051115