Improved FFT Approximations of Probability Functions Based on Modified Quadrature Rules

In the last ten years, the traditional Simpson quadrature rule for numerical integration has been improved to an optimal 3-point quadrature formula and a modified Simpson rule that takes additionally the first derivative of the approximated integrand at the two end-points of integration into account. The impact of the improved quadrature rules on the fast Fourier transform (FFT) approximation of probability density functions with known characteristic functions is discussed. The quality of the approximations is measured with a discrete version of the total variation distance. Numerical examples suggest that a discrete total variation distance of approximately 0.25 is the best possible attainable value across all the considered FFT approximations. Mathematics Subject Classification: 41A55, 62E17, 65C60, 65T50, 91G60