cf_RectangularSymmetric: 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 RECTANGULAR random variables

Description

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

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

The characteristic function of X ~ RectangularSymmetric is defined by

cf(t) = cf_RectangularSymmetric(t) = sinc(t) = sin(t)/t.

Usage

cf_RectangularSymmetric(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 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 RECTANGULAR random variables.

Note

Ver.: 16-Sep-2018 18:38:11 (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/Uniform_distribution_(continuous).

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_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_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_BetaSymmetric(), cf_TSPSymmetric(), cf_TrapezoidalSymmetric()

Examples

## EXAMPLE 1
# CF of the Rectangular distribution on (-1,1)
t <- seq(from = -50,
         to = 50,
         length.out = 201)
plotReIm(function(t)
        cf_RectangularSymmetric(t),
        t,
        title = "CF of the Rectangular distribution on (-1,1)")

## EXAMPLE 2
# CF of a weighted linear combination of independent Rectangular RVs
t <- seq(from = -10,
         to = 10,
         length.out = 201)
coef <- c(1, 2, 3, 4, 5) / 15
plotReIm(function(t)
        cf_RectangularSymmetric(t, coef),
        t,
        title = "CF of a weighted linear combination of Rectangular RVs")

## EXAMPLE 3
# PDF/CDF of a weighted linear combination of independent Rectangular RVs
coef <- c(1, 2, 3, 4, 5) / 15
cf <- function(t)
        cf_RectangularSymmetric(t, 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.