cf_TSPSymmetric: Characteristic function of a linear combination (resp....

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

Characteristic function of a linear combination (resp. convolution) of independent symmetric (location and scale shifted) TWO-SIDED-POWER (TSP) random variables.

Description

That is, cfS_TSPSymmetric(t, theta, mu, sigma, coef, niid) evaluates the characteristic function cf(t) of Y = sum_{i=1}^N coef_i * (mu_i + sigma_i * X_i), where X_i ~ TSP(theta_i) are inedependent RVs, with symmetric TSP distributions defined on the interval (-1,1) with zero mean and variance Var(X_i) = 2*theta_i*gamma(theta_i)/gamma(3+theta_i), where theta_i > 0 are shape parameters for all zeqni = 1,...,N.

The characteristic function of the random variablemu + sigma*X, where X ~ TSP(theta) is given by cf(t) = cfS_TSPSymmetric(t,theta,mu,sigma) = 1/2 * exp(1i*t*mu) * (hypergeom1F1(1,1+theta,1i*t*sigma) + hypergeom1F1(1,1+theta,-1i*t*sigma)).

Hence, the characteristic function of Y = coef_1*(mu_1+sigma_1*X_1) + coef_N*(mu_N+sigma_N*X_N) is cf_Y(t) = exp(1i*mu*t) *(cf_1(coef_1*sigma_1*t) * cf_N(coef_N*sigma_N*t)), where cf_i(t) is the characteristic function of X_i ~ TSP(theta_i).

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_TSPSymmetric(t, theta, mu, sigma, coef, niid)

Arguments

t	vector or array of real values, where the CF is evaluated.
theta	vector of the shape parameters theta > 0. If theta is scalar, it is assumed that all parameters theta are equal. If empty, default value is theta = 1.
mu	vector of location parameters, mu in Real. If empty, default value is mu = 0.
sigma	vector of scale parameters, sigma_i > 0. If empty, default value is sigma = 1.
coef	vector of the coefficients of the linear combination of the log-transformed 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 n, 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 of a linear combination (resp. convolution) of independent symmetric (location and scale shifted) TWO-SIDED-POWER (TSP) random variables.

Note

Ver.: 11-Aug-2021 16:18:40 (consistent with Matlab CharFunTool v1.5.1, 24-Jun-2017 18:25:56).

References

[1] 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. [2] VAN DORP, R.J., KOTZ, S. (2003). Generalizations of two-sided power distributions and their convolution. Communications in Statistics-Theory and Methods, 32(9), 1703-1723.

See Also

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_RectangularSymmetric(), cf_Student(), 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_RectangularSymmetric(), cf_TrapezoidalSymmetric()

Examples

## EXAMPLE 1
# CF of the symmetric TSP distribution with theta =3/2 on (-1,1)
theta <- 3/2
t <- seq(from = -50,
         to = 50,
         length.out =501)
plotReIm(function(t)
        cf_TSPSymmetric(t, theta),
        t,
        title = "CF of the symmetric TSP distribution  on (-1,1)")



##EXAMPLE2
# PDF/CDF of the symmetric TSP distribution on (-1,1)
thet <- 3/2
cf <- function(t)
        cf_TSPSymmetric(t, theta)
x <- seq(-1,1,length.out = 101)
xRange <- 2
options <- list()
options.N <-2^8
options.dt <- 2*pi/xRange
result <- cf2DistGP(cf, x, c(), options)

##EXAMPLE 3
# CF of the weighted linear combination of TSP RVs
theta <- c(1, 2, 3, 4, 5)/2
mu <- c(1, 2, 0, 0, 0)
sigma <- c(1, 2, 3, 4, 5)/5
coef <- 1/5
t <- seq(-50,50, length.out=  501)
plotReIm(function(t)
        cf_TSPSymmetric(t, theta, mu, sigma, coef),
        t,
        title = "CF of the weighted linear combination of TSP RVs")

## EXAMPLE 4
# CDF/PDF of the weighted linear combination of TSP RVs
thet <- c(1, 2, 3, 4, 5)/2
mu <- 0
sigma <- c(5, 4, 3, 2, 1)
coef <- 1/5
t <-seq(-50, 50, length.out = 501)
cf <- function(t)
        cf_TSPSymmetric(t, theta, mu, sigma, coef)
prob <- c(0.9, 0.95, 0.99)
options <- list()
options.N = 2^12
result <- cf2DistGP(cf, c(), prob, options)

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