cf_BetaNC: 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 non-central BETA random variables

Description

cf_BetaNC(t, alpha, beta, delta, coef, niid, tol, type) evaluates characteristic function of a linear combination (resp. convolution) of independent non-central BETA random variables (Type I and Type II), with distributions Beta(\alpha_i,\beta_i,\delta_i) defined on the interval (0,1), and specified by the parameters \alpha_i, \beta_i, and the noncentrality parameters delta_i.

The noncentral beta distribution has two types. The Type I is the distribution of the random variable B_1 = X_1/(X_1+X_2), X1 ~ Gamma(\alpha, \gamma, \delta) and X_2 ~ Gamma(\beta, \gamma). The Type II noncentral beta distribution is the distribution of the ratio random variable B_2 = X_1/(X_1+X_2), where X_1 ~ Gamma(\alpha,\gamma) and X_2 ~ Gamma(\beta, \gamma, \delta).

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

For Type I noncentral beta distribution, the characteristic function of X ~ Beta(\alpha, \beta, \delta) is Poisson mixture of the central Beta CFs of the form

cf(t) = cf_BetaNC(t, \alpha, \beta, \delta) = exp(-\delta/2) sum_{j=1}^Inf (\delta/2)^j/j! * cf_beta(\alpha+j, \beta)

where cf_beta(\alpha+j, \beta) are the CFs of central Beta RVs with parameters alpha+j and beta.

For Type I noncentral beta distribution, the characteristic function of X ~ Beta(\alpha, \beta, \delta) is Poisson mixture of the central Beta CFs of the form

cf(t) = cf_BetaNC(t, \alpha, \beta, \delta) = exp(-\delta/2) sum_{j=1}^Inf (\delta/2)^j/j! * cf_beta(\alpha, \beta+j)

where cf_beta(\alpha, \beta+j) are the CFs of central Beta RVs with parameters \alpha+j and \beta.

Hence,the characteristic function of Y = coef(1)*X_1 + ... + coef(N)*X_N is

cf(t) = cf_X_1(coef(1)*t) * ... * cf_X_N(coef(N)*t),

where X_i ~ BetaNC(alpha(i),beta(i),delta(i)) with cf_X_i(t).

Usage

cf_BetaNC(t, alpha, beta, delta, coef, niid, tol, type)

Arguments

t	vector or array of real values, where the CF is evaluated.
alpha	vector of the 'shape' parameters alpha > 0. If empty, default value is alpha = 1.
beta	vector of the 'shape' parameters beta > 0. If empty, default value is beta = 1.
delta	vector of the non-centrality parameters delta > 0. If empty, default value is delta = 0.
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.
tol	tolerance factor for selecting the Poisson weights, i.e. such that PoissProb > tol. If empty, default value is tol = 1e-12.
type	indicator of the type of the noncentral distribution (Type I = 1 or Type II = 2). If empty, default value is type = 1.

Value

Characteristic function cf(t) of a linear combination of independent non-central BETA random variables.

Note

Ver.: 19-Sep-2018 23:51:40 (consistent with Matlab CharFunTool v1.3.0, 07-Feb-2018 13:53:40).

See Also

For more details see WIKIPEDIA: https://en.wikipedia.org/wiki/Noncentral_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_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 Non-central Probability Distribution: cf_FisherSnedecorNC(), cf_LogRV_BetaNC(), cf_LogRV_ChiSquareNC(), cf_LogRV_FisherSnedecorNC()

Examples

## EXAMPLE 1
# CF of non-central Beta RV with delta = 1
alpha <- 1
beta  <- 3
delta <- 1
t <- seq(from = -50,
         to = 50,
         length.out = 201)
functions_to_plot <- list(function(t) cf_BetaNC(t, alpha, beta, delta, type = 1),
                          function(t) cf_BetaNC(t, alpha, beta, delta, type = 2))
plotReIm2(functions_to_plot, list(t, t), title = 'CF of Type I and II Beta RV')

## EXAMPLE 2
# CDF/PDF of non-central Beta RV with delta = 1
alpha <- 1
beta  <- 3
delta <- 1
cf <- function(t)
        cf_BetaNC(t, alpha, beta, delta)
options <- list()
options$xMin <- 0
options$xMax <- 1
result <- cf2DistGP(cf = cf, options = options)

## EXAMPLE 3
# CDF/PDF of the linear combination of non-central Beta RVs
alpha <- c(5, 4, 3)
beta  <- c(3, 4, 5)
delta <- c(0, 1, 2)
coef  <- 1 / 3
cf <- function(t)
        cf_BetaNC(t, alpha, beta, delta, coef)
options <- list()
options$xMin <- 0
options$xMax <- 1
result <- cf2DistGP(cf = cf, options = options)

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