cf2DistGP: Evaluating CDF/PDF/QF from CF of a continous distribution F...

In gajdosandrej/CharFunToolR: Numerical Computation Cumulative Distribution Function and Probability Density Function from Characteristic Function

Evaluating CDF/PDF/QF from CF of a continous distribution F by using the Gil-Pelaez inversion formulae.

Description

cf2DistGP(cf, x, prob, options) calcuates the CDF/PDF/QF from the Characteristic Function CF by using the Gil-Pelaez inversion formulae:

cdf(x) = 1/2 + (1/\pi) * Integral_0^inf imag(exp(-1i*t*x)*cf(t)/t)*dt,

pdf(x) = (1/\pi) * Integral_0^inf real(exp(-1i*t*x)*cf(t))*dt.

The FOURIER INTEGRALs are calculated by using the simple TRAPEZOIDAL QUADRATURE method, see below.

Usage

cf2DistGP(cf, x, prob, options)

Arguments

cf	function handle for the characteristic function CF.
x	vector of values where the CDF/PDF is computed.
prob	vector of values from [0,1] for which the quantile function is evaluated.
options	structure (list) with the following default parameters: options$isCompound = FALSE treat the compound distributions, of the RV Y = X_1 + ... + X_N, where N is discrete RV and X\ge0 are iid RVs from nonnegative continuous distribution, options$isCircular = FALSE treat the circular distributions on (-\pi, \pi), options$isInterp = FALSE create and use the interpolant functions for PDF/CDF/RND, options$N = 2^10 N points used by FFT, options$xMin = -Inf set the lower limit of x, options$xMax = Inf set the upper limit of x, options$xMean = NULL set the MEAN value of x, options$xStd = NULL set the STD value of x, options$dt = NULL set grid step dt = 2*\pi/xRange, options$T = NULL set upper limit of (0,T), T = N*dt, options$SixSigmaRule = 6 set the rule for computing domain, options$tolDiff = 1e-4 tol for numerical differentiation, options$isPlot = TRUE plot the graphs of PDF/CDF, options$DIST list with information for future evaluations, options$DIST is created automatically after first call: options$DIST$xMin the lower limit of x, options$DIST$xMax the upper limit of x, options$DIST$xMean the MEAN value of x, options$DIST$cft CF evaluated at t_j : cf(t_j).

Details

If options.DIST is provided, then cf2DistGP evaluates CDF/PDF based on this information only (it is useful, e.g., for subsequent evaluation of the quantiles). options.DIST is created automatically after first call. This is supposed to be much faster, bacause there is no need for further evaluations of the characteristic function. In fact, in such case the function handle of the CF is not required, i.e. in such case do not specify cf.

The required integrals are evaluated approximately by using the simple trapezoidal rule on the interval (0,T), where T = N * dt is a sufficienly large integration upper limit in the frequency domain.

If the optimum values of N and T are unknown, we suggest, as a simple rule of thumb, to start with the application of the six-sigma-rule for determining the value of dt = (2*\pi)/(xMax-xMin), where xMax = xMean + 6*xStd, and xMin = xMean - 6*xStd, see [1].

Please note that THIS (TRAPEZOIDAL) METHOD IS AN APPROXIMATE METHOD: Frequently, with relatively low numerical precision of the results of the calculated PDF/CDF/QF, but frequently more efficient and more precise than comparable Monte Carlo methods.

However, the numerical error (truncation error and/or the integration error) could be and should be properly controled!

CONTROLING THE PRECISION: Simple criterion for controling numerical precision is as follows: Set N and T = N*dt such that the value of the integrand function Imag(e^(-1i*t*x) * cf(t)/t) is sufficiently small for all t > T, i.e. PrecisionCrit = abs(cf(t)/t) <= tol, for pre-selected small tolerance value, say tol = 10^-8. If this criterion is not valid, the numerical precission of the result is violated, and the method should be improved (e.g. by selecting larger N or considering other more sofisticated algorithm - not considered here).

Value

result

structure (list) with CDF/PDF/QF amd further details.

Note

Ver.: 16-Sep-2018 17:59:07 (consistent with Matlab CharFunTool v1.3.0, 22-Sep-2017 11:11:11).

References

[1] WITKOVSKY, V.: On the exact computation of the density and of the quantiles of linear combinations of t and F random variables. Journal of Statistical Planning and Inference, 2001, 94, 1-13.

[2] WITKOVSKY, V.: Matlab algorithm TDIST: The distribution of a linear combination of Student's t random variables. In COMPSTAT 2004 Symposium (2004), J. Antoch, Ed., Physica-Verlag/Springer 2004, Heidelberg, Germany, pp. 1995-2002.

[3] WITKOVSKY, V., WIMMER, G., DUBY, T. Logarithmic Lambert W x F random variables for the family of chi-squared distributions and their applications. Statistics & Probability Letters, 2015, 96, 223-231.

[4] WITKOVSKY, V.: Numerical inversion of a characteristic function: An alternative tool to form the probability distribution of output quantity in linear measurement models. Acta IMEKO, 2016, 5(3), 32-44.

[5] WITKOVSKY, V., WIMMER, G., DUBY, T. Computing the aggregate loss distribution based on numerical inversion of the compound empirical characteristic function of frequency and severity. ArXiv preprint, 2017, arXiv:1701.08299.

See Also

For more details see: https://arxiv.org/pdf/1701.08299.pdf.

Other CF Inversion Algorithm: cf2DistFFT(), cf2PMF_FFT()

Examples

## EXAMPLE1
# Calculate CDF/PDF of N(0,1) by inverting its CF
cf <- function(t)
        exp(-t ^ 2 / 2)
result <- cf2DistGP(cf)

## EXAMPLE2
# PDF/CDF of the compound Binomial-Exponential distribution
n <- 25
p <- 0.3
lambda <- 5
cfX <- function(t)
        cfX_Exponential(t, lambda)
cf <- function(t)
        cfN_Binomial(t, n, p, cfX)
x <- seq(0, 5, length.out = 101)
prob <- c(0.9, 0.95, 0.99)
options <- list()
options$isCompound <- TRUE
result <- cf2DistGP(cf, x, prob, options)

## EXAMPLE3
# PDF/CDF of the compound Poisson-Exponential distribution
lambda1 <- 10
lambda2 <- 5
cfX <- function(t)
        cfX_Exponential(t, lambda2)
cf <- function(t)
        cfN_Poisson(t, lambda1, cfX)
x <- seq(0, 8, length.out = 101)
prob <- c(0.9, 0.95, 0.99)
options <- list()
options$isCompound <- TRUE
result <- cf2DistGP(cf, x, prob, options)

gajdosandrej/CharFunToolR documentation built on June 3, 2024, 7:46 p.m.