The Fourier-Series Method For Inverting Transforms Of Probability Distributions

The Fourier-Series Method For Inverting Transforms Of Probability Distributions (1991)

Cached

Download Links

[www.research.att.com]

Other Repositories/Bibliography

DBLP

by Joseph Abate , Ward Whitt

Citations:

135 - 50 self

Versions

Version History

Metadata Version 5

User correction supplied by mph

Datum	Value	Source
TITLE	The Fourier-Series Method For Inverting Transforms Of Probability Distributions	user correction - Legacy Corrections
AUTHOR NAME	Joseph Abate	user correction
AUTHOR AFFIL	900 Hammond Road	user correction
AUTHOR ADDR	AT&T Bell Laboratories; Ridgewood, NJ 07450-2908 Room 2C-178; Murray Hill, NJ 07974-0636	user correction
AUTHOR NAME	Ward Whitt	user correction
AUTHOR AFFIL	900 Hammond Road	user correction
AUTHOR ADDR	AT&T Bell Laboratories; Ridgewood, NJ 07450-2908 Room 2C-178; Murray Hill, NJ 07974-0636	user correction
ABSTRACT	This paper reviews the Fourier-series method for calculating cumulative distribution functions (cdf's) and probability mass functions (pmf's) by numerically inverting characteristic functions, Laplace transforms and generating functions. Some variants of the Fourier-series method are remarkably easy to use, requiring programs of less than fifty lines. The Fourier-series method can be interpreted as numerically integrating a standard inversion integral by means of the trapezoidal rule. The same formula is obtained by using the Fourier series of an associated periodic function constructed by aliasing; this explains the name of the method. This Fourier analysis applies to the inversion problem because the Fourier coefficients are just values of the transform. The mathematical centerpiece of the Fourier-series method is the Poisson summation formula, which identifies the discretization error associated with the trapezoidal rule and thus helps bound it. The greatest difficulty is approximately calculating the infinite series obtained from the inversion integral. Within this framework, lattice cdf's can be calculated from generating functions by finite sums without truncation. For other cdf's, an appropriate truncation of the infinite series can be determined from the transform based on estimates or bounds. For Laplace transforms, the numerical integration can be made to produce a nearly alternating series, so that the convergence can be accelerated by techniques such as Euler summation. Alternatively, the cdf can be perturbed slightly by convolution smoothing or windowing to produce a truncation error bound independent of the original cdf. Although error bounds can be determined, an effective approach is to use two different methods without elaborate error analysis. For this...	user correction - Legacy Corrections
YEAR	1991	user correction - Legacy Corrections
CITATIONS	124 found	ParsCit 1.0