cf_ArcsineSymmetric: Characteristic function of a linear combination of...

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

Characteristic function of a linear combination of independent zero-mean symmetric ARCSINE random variables

Description

cfS_ArcsineSymmetric(t, coef, niid) evaluates Characteristic function of a linear combination (resp. convolution) of independent zero-mean symmetric ARCSINE random variables defined on the interval (-1,1).

That is, cfS_ArcsineSymmetric evaluates the characteristic function cf(t) of Y = sum_{i=1}^N coef_i * X_i, where X_i ~ ArcsineSymmetric are independent RVs defined on (-1,1), for all i = 1,...,N.

The characteristic function of X ~ ArcsineSymmetric is defined by

cf(t) = cf_ArcsineSymmetric(t) = besselj(0,t).

Usage

cf_ArcsineSymmetric(t, coef, niid)

Arguments

t	vector or array of real values, where the CF is evaluated.
coef	vector of the coefficients of the linear combination of the symmetric Arcsinne distributed random variables. If coef is scalar, it is assumed that all coefficients are equal. If empty, default value is coef = 1.
niid	scalar convolution coeficient niid, such that Z = Y + ... + Y is sum of niid iid random variables Y, where each Y = sum_{i=1}^N coef(i) * log(X_i) is independently and identically distributed random variable. If empty, default value is niid = 1.

Value

Characteristic function cf(t) of a linear combination of independent zero-mean symmetric ARCSINE random variables.

Note

Ver.: 16-Sep-2018 18:02:14 (consistent with Matlab CharFunTool v1.3.0, 02-Jun-2017 12:08:24).

References

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

See Also

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

Other Continuous Probability Distribution: cfS_Arcsine(), cfS_Beta(), cfS_Gaussian(), cfS_Laplace(), cfS_Rectangular(), cfS_Student(), cfS_TSP(), cfS_Trapezoidal(), cfS_Triangular(), cfS_Wigner(), cfX_ChiSquare(), cfX_Exponential(), cfX_FisherSnedecor(), cfX_Gamma(), cfX_InverseGamma(), cfX_LogNormal(), cf_BetaNC(), cf_BetaSymmetric(), cf_Beta(), cf_ChiSquare(), cf_Exponential(), cf_FisherSnedecorNC(), cf_FisherSnedecor(), cf_Gamma(), cf_InverseGamma(), cf_Laplace(), cf_LogRV_BetaNC(), cf_LogRV_Beta(), cf_LogRV_ChiSquareNC(), cf_LogRV_ChiSquare(), cf_LogRV_FisherSnedecorNC(), cf_LogRV_FisherSnedecor(), cf_LogRV_MeansRatioW(), cf_LogRV_MeansRatio(), cf_LogRV_WilksLambdaNC(), cf_LogRV_WilksLambda(), cf_Normal(), cf_RectangularSymmetric(), cf_Student(), cf_TSPSymmetric(), cf_TrapezoidalSymmetric(), cf_TriangularSymmetric(), cf_vonMises()

Other Symmetric Probability Distribution: cfS_Arcsine(), cfS_Beta(), cfS_Gaussian(), cfS_Laplace(), cfS_Rectangular(), cfS_Student(), cfS_Trapezoidal(), cf_BetaSymmetric(), cf_RectangularSymmetric(), cf_TSPSymmetric(), cf_TrapezoidalSymmetric()

Examples

## EXAMPLE 1
# CF of the symmetric Arcsine distribution on (-1,1)
t <- seq(from = -50,
         to = 50,
         length.out = 501)
plotReIm(function(t)
        cf_ArcsineSymmetric(t),
        t,
        title = "CF of the the Arcsine distribution on (-1,1)")

## EXAMPLE 2
# CF of a linear combination of independent Arcsine RVs
t <- seq(from = -1,
         to = 1,
         length.out = 501)
coef <- c(1, 2, 3, 4, 5)
plotReIm(function(t)
        cf_ArcsineSymmetric(t, coef),
        t,
        title = "CF of a linear combination of independent Arcsine RVs")

## EXAMPLE 3
## PDF/CDF of a linear combination of independent Arcsine RVs
coef <- c(1, 2, 3, 4, 5)
cf   <- function(t)
        cf_ArcsineSymmetric(t, coef)
x    <- seq(from = -20,
            to = 20,
            length.out = 201)
prob <- c(0.9, 0.95, 0.99)
options <- list()
options$N <- 2 ^ 12
result <- cf2DistGP(cf, x, prob, options)

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