Chebyshev approximation of the distribution function of nonnegative random variables
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.
scientific article; zbMATH DE number 1051115
scientific article