CharFun (version 0.1.0)

cf2DistGP: Evaluating CDF/PDF/QF from CF of a continous distribution F by using the Gil-Pelaez inversion formulae.

Description

cf2DistGP(cf, x, prob, options) evaluates the CDF/PDF/QF (Cumulative Distribution Function, Probability Density Function, Quantile Function) 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.

Usage

cf2DistGP(cf, x, prob, option, isCompound, isCircular, N, SixSigmaRule, xMin,
  xMax, isPlot)

Arguments

cf

function handle for the characteristic function CF

x

vector of values from domain of the distribution F, if missing, cf2DistFFT automatically selects vector x from the domain

prob

vector of values from [0,1] for which the quantiles will be estimated

option

list with the following (default) parameters:

• option$isCompound = FALSE treat the compound distributions
• option$isCircular = FALSE treat the circular distributions
• option$isInterp = FALSE create and use the interpolant functions for PDF/CDF/RND
• option$N = 2^10 set N points used by FFT
• option$xMin = -Inf set the lower limit of X
• option$xMax = Inf set the upper limit of X
• option$xMean = NULL set the MEAN value of X
• option$xStd = NULL set the STD value of X
• option$dt = NULL set grid step dt = 2*pi/xRange
• option$T = NULL set upper limit of (0,T), T = N*dt
• option$SixSigmaRule = 6 set the rule for computing domain
• option$tolDiff = 1e-4 tol for numerical differentiation
• option$isPlot = TRUE plot the graphs of PDF/CDF

• option$DIST list with information for future evaluations. option$DIST is created automatically after first call:
  • option$DIST$xMean
  • option$DIST$xMin
  • option$DIST$xMax
  • option$DIST$xRange
  • option$DIST$cft
  • option$DIST$N
  • option$DIST$k
  • option$DIST$t
  • option$DIST$dt

isCompound

treat the compound distributions, default FALSE

isCircular

treat the circular distributions, default FALSE

N

N points used by GP, default 2^10 (2^14 for compound distribution)

SixSigmaRule

set the rule for computing domain, default 6

xMin

set the lower limit of X

xMax

set the upper limit of X

isPlot

plot the graphs of PDF/CDF, default TRUE

Value

result - list with the following results values:

result$x

<- x

result$cdf

<- cdf (Cumulative Distribution Function)

result$pdf

<- pdf (Probability Density Function)

result$prob

<- prob

result$qf

<- qf (Quantile Function)

result$SixSigmaRule

<- option$SixSigmaRule

result$N

<- option$N

result$dt

<- dt

result$T

<- t[(length(t))]

result$PrecisionCrit

<- PrecisionCrit

result$myPrecisionCrit

<- options$crit

result$isPrecisionOK

<- isPrecisionOK

result$xMean

<- xMean

result$xRange

<- xRange

result$xSTD

<- xStd

result$xMin

<- xMin

result$xMAx

<- xMax

result$cf

<- cf

result$const

<- const

result$isCompound

<- option$isCompound

result$details$count

<- count

result$details$correction

<- correction

result$option

<- option

result$tictoc

<- toc

See Also

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

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)
result <- cf2DistGP(cf, x, prob, isCompound = TRUE)

## 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)
result <- cf2DistGP(cf, x, prob, isCompound = TRUE)