Accession Number : ADA330541

Title :   Evaluation of Small Tail Probabilities Directly from the Characteristic Function

Descriptive Note : Progress rept.

Corporate Author : NAVAL UNDERSEA WARFARE CENTER NEWPORT DIV RI

Personal Author(s) : Nuttall, Albert H.

PDF Url : ADA330541

Report Date : 15 SEP 1997

Pagination or Media Count : 53

Abstract : An efficient, fast, and accurate Fourier transform technique for obtaining small tail probabilities for both the probability density function and the exceedance distribution function, directly from the characteristic function, is derived and demonstrated numerically for several examples. The method is especially useful when analytic or asymptotic expressions for the probabilities are unavailable or unknown. By choosing the shift parameter r close to the highest singularity of the characteristic function in the complex plane, very small values of the tail probabilities of the density function and exceedance distribution function can be realized. The cost in this approach is that the sampling increment must then be taken small, in order to avoid aliasing. Finer sampling necessitates more computer time and effort, but it does not require more storage; rather, prealiasing can be advantageously employed to keep the fast Fourier transform size at reasonable values. The fast Fourier transform size has no effect upon the errors caused by aliasing and truncation; rather, the size merely controls the spacing at which the probability density function and exceedance distribution function outputs are calculated. Tail probabilities in the E-50 range are readily available with a computer limited to 15 significant decimal digits.

Descriptors :   *PROBABILITY DENSITY FUNCTIONS, *FAST FOURIER TRANSFORMS, RANDOM VARIABLES, STATISTICAL SAMPLES, ASYMPTOTIC SERIES, NORMAL DENSITY FUNCTIONS.

Subject Categories : Operations Research

Distribution Statement : APPROVED FOR PUBLIC RELEASE