Accurate Efficient Evaluation of Cumulative or Exceedance Probability Distributions Directly from Characteristic Functions

Abstract

An accurate and efficient method of evaluating the entire cumulative or exceedance probability distribution, via one fast Fourier transform of the sampled characteristic function, is presented. The sampling rate applied to the characteristic function results in aliasing of the probability density function, while the limited extent of the sampling gives rise to a systematic disturbance in the calculated probability distribution. Both types of errors are easily recognizable and can be controlled by a trail and error procedure whereby the calculated distributions are plotted for observation and modification. The size of the fast Fourier transform determines the number of distribution values available, but has no effect upon the acccuracy of the result. Regardless of the number of characteristic function evaluations required for accurate results, the storage required is just that corresponding to the size of the fast Fourier transform. A program for the procedure is presented, and the inputs required of the user are indicated. Several representative examples and plots illustrate the utility of the approach.