cf_BetaSymmetric: 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 BETA random variables

Description

cf_BetaSymmetric(t, theta, coef, niid) evaluates the characteristic function of a linear combination (resp. convolution) of independent zero-mean symmetric BETA random variables defined on the interval (-1,1).

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

The characteristic function of X ~ BetaSymmetric(\theta) is defined by

cf(t) = cf_BetaSymmetric(t,\theta) = gamma(1/2+\theta) * (t/2)^(1/2-\theta) * besselj(\theta-1/2,t).

SPECIAL CASES:
1) \theta = 1/2; Arcsine distribution on (-1,1): cf(t) = besselj(0,t),
2) \theta = 1; Rectangular distribution on (-1,1): cf(t) = sin(t)/t.

Usage

cf_BetaSymmetric(t, theta, coef, niid)

Arguments

t	vector or array of real values, where the CF is evaluated.
theta	the 'shape' parameter theta > 0. If empty, default value is theta = 1.
coef	vector of the coefficients of the linear combination of Beta 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 BETA random variables.

Note

Ver.: 16-Sep-2018 18:10:34 (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/Beta_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_ArcsineSymmetric(), cf_BetaNC(), 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_ArcsineSymmetric(), cf_RectangularSymmetric(), cf_TSPSymmetric(), cf_TrapezoidalSymmetric()

Examples

## EXAMPLE 1
# CF of the symmetric Beta distribution with theta = 3/2 on (-1,1)
theta <- 3 / 2
t <- seq(from = -50,
         to = 50,
         length.out = 201)
plotReIm(function(t)
        cf_BetaSymmetric(t, theta),
        t,
        title = "CF of the symmetric Beta distribution on (-1,1)")

## EXAMPLE 2
# CF of a linear combination of independent Beta RVs
t <- seq(from = -20,
         to = 20,
         length.out = 201)
theta <- c(3, 3, 4, 4, 5) / 2
coef <- c(1, 2, 3, 4, 5) / 15
plotReIm(function(t)
        cf_BetaSymmetric(t, theta, coef),
        t,
        title = "CF of a linear combination of independent Beta RVs")

## EXAMPLE 3
# PDF/CDF of a weighted linear combination of independent Beta RVs
theta <- c(3, 3, 4, 4, 5) / 2
coef <- c(1, 2, 3, 4, 5) / 15
cf   <- function(t)
        cf_BetaSymmetric(t, theta, coef)
x <- seq(from = -1,
         to = 1,
         length.out = 201)
prob <- c(0.9, 0.95, 0.99)
options <- list()
options$N <- 2 ^ 12
options$xMin <- -1
options$xMax <- 1
result <- cf2DistGP(cf, x, prob, options)

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