cf_Gamma: 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 GAMMA random variables

Description

cf_Gamma(t, alpha, beta, coef, niid) evaluates the characteristic function of a linear combination (resp. convolution) of independent GAMMA random variables.

That is, cf_Gamma evaluates the characteristic function cf(t) of Y =sum_{i=1}^N coef_i * X_i, where X_i ~ Gamma(\alpha_i,\beta_i) are inedependent RVs, with the shape parameters \alpha_i > 0 and the rate parameters \beta_i > 0, for i = 1,...,N.

The characteristic function of Y is defined by

cf(t) = Prod( (1 - i*t*coef(i)/\beta(i))^(-\alpha(i)) ).

Usage

cf_Gamma(t, alpha, beta, coef, niid)

Arguments

t	vector or array of real values, where the CF is evaluated.
alpha	the shape parameter alpha > 0. If empty, default value is alpha = 1.
beta	the rate (1/scale) parameter beta > 0. If empty, default value is beta = 1.
coef	- vector of the coefficients of the linear combination of the GAMMA 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.

Details

PARAMETRIZATION:
Notice that there are three different parametrizations for GAMMA distribution in common use:
i) With a shape parameter k and a scale parameter theta.
ii) With a shape parameter \alpha = k and an inverse scale parameter \beta = 1/\theta, called a rate parameter.
iii) With a shape parameter k and a mean parameter \mu = k/\beta. In each of these three forms, both parameters are positive real numbers.

Here, cf_Gamma implements the shape-rate parametrization with parameters alpha and beta, respectively.

SPECIAL CASES:
1) If X ~ Gamma(1,\lambda) (shape-rate parametrization), then X has an exponential distribution with rate parameter lambda. 2) If X ~ Gamma(df/2,1/2)(shape-rate parametrization), then X ~ ChiSquared(df), the chi-squared distribution with df degrees offreedom. Conversely, if Q ~ ChiSquared(df) and c is a positive constant, then cQ ~ Gamma(df/2,1/2c). 3) If X ~ Gamma(\alpha,\theta) and Y ~ Gamma(\beta,\theta) are independently distributed, then X/(X + Y) has a beta distribution with parameters \alpha and \beta.

Value

Characteristic function cf(t) of a linear combination of independent GAMMA random variables.

Note

Ver.: 16-Sep-2018 18:26:34 (consistent with Matlab CharFunTool v1.3.0, 10-May-2017 18:11:50).

See Also

For more details see WIKIPEDIA: https://en.wikipedia.org/wiki/Gamma_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_BetaSymmetric(), cf_Beta(), cf_ChiSquare(), cf_Exponential(), cf_FisherSnedecorNC(), cf_FisherSnedecor(), 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()

Examples

## EXAMPLE 1
# CF of a linear combination of independent Gamma RVs
coef <- 1 / (((1:50) - 0.5) * pi) ^ 2
plot(
        1:50,
        coef,
        xlab = "",
        ylab = "",
        type = "p",
        pch = 20,
        col = "blue",
        cex = 1,
        main = expression('Coefficients of the linear combination of GAMMA RVs')
)
lines(1:50, coef, col = "blue")
alpha <- 5 / 2
beta <- 1 / 2
t <- seq(from = -10,
         to = 10,
         length.out = 201)
plotReIm(function(t)
        cf_Gamma(t, alpha, beta, coef),
        t,
        title = "Characteristic function of the linear combination of GAMMA RVs")

## EXAMPLE 2
# PDF/CDF from the CF by cf2DistGP
alpha <- 5 / 2
beta <- 1 / 2
coef <- 1 / (((1:50) - 0.5) * pi) ^ 2
cf <- function(t)
        cf_Gamma(t, alpha, beta, coef)
options <- list()
options$N <- 2 ^ 10
options$xMin <- 0
result <- cf2DistGP(cf = cf, options = options)

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