cfX_PearsonV: Characteristic function of Pearson type V distribution

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

Characteristic function of Pearson type V distribution

Description

cfX_PearsonV(t, alpha, beta) evaluates the characteristic function of the Pearson type V distribution with the parameters alpha (shape, alpha > 0) and beta (scale, beta > 0), computed for real vector argument t, i.e.

cfX_PearsonV(t, \alpha, \beta) = (2/gamma(\alpha)) * (-1i*t/\beta)^(\alpha/2) * besselk(\alpha,2*sqrt(-1i*t/\beta)),

where besselk(a,z) denotes the modified Bessel function of the second order.

Usage

cfX_PearsonV(t, alpha = 1, beta = 1)

Arguments

t	vector of real values where the CF is evaluated, i.e. CF(t).
alpha	scalar shape parameter of the Pearson VI distribution, alpha > 0.
beta	scalar scale parameter of the Pearson VI distribution, beta > 0.

Value

Characteristic function cf(t) of the Pearson type V distribution.

Note

Ver.: 16-Sep-2018 19:28:37 (consistent with Matlab CharFunTool v1.3.0, 15-Nov-2016 13:36:26).

See Also

For more details see WIKIPEDIA: https://en.wikipedia.org/wiki/Pearson_distribution.

Examples

## EXAMPLE 1
# CF of the PearsonV distribution
alpha <- 3 / 2
beta <- 2 / 3
t <- seq(-10, 10, length.out = 1001)
plotReIm(function(t)
        cfX_PearsonV(t, alpha, beta), t,
        title = "CF of the PearsonV distribution with alpha = 3/2, beta = 2/3")

## EXAMPLE 2
# PDF/CDF of the Beta distribution with alpha = 3/2, beta = 2/3
alpha <- 3 / 2
beta <- 2 / 3
prob <- c(0.9, 0.95, 0.99)
x <- seq(0, 40, length.out = 101)
cf <- function(t)
        cfX_PearsonV(t, alpha, beta)
options <- list()
options$xMin <- 0
options$N <- 2 ^ 10
options$SixSigmaRule <- 10
result <- cf2DistGP(cf, x, prob, options)

## EXAMPLE 3
# PDF/CDF of the compound Binomial-PearsonV distribution
n <- 25
p <- 0.3
alpha <- 3 / 2
beta <- 2 / 3
prob <- c(0.9, 0.95, 0.99)
x <- seq(0, 200, length.out = 101)
cfX <- function(t)
        cfX_PearsonV(t, alpha, beta)
cf <- function(t)
        cfN_Binomial(t, n, p, cfX)
options <- list()
options$isCompound <- TRUE
options$N <- 2 ^ 10
result <- cf2DistGP(cf, x, prob, options)

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