cf_LogRV_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 LOG-TRANSFORMED non-central BETA random variables

Description

cf_LogRV_BetaNC(t, alpha, beta, delta, coef, niid, tol, type) evaluates characteristic function of a linear combination (resp. convolution) of independent LOG-TRANSFORMED non-central BETA random variables, (Type I and Type II), with their distributions 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 B1 = X1/(X1+X2), X1 ~ Gamma(\alpha, \gamma, \delta) and X2 ~ Gamma(\beta, \gamma). The Type II noncentral beta distribution is the distribution of the ratio random variable B2 = X1/(X1+X2), where X1 ~ Gamma(\alpha, \gamma) and X2 ~ Gamma(\beta, \gamma, \delta).

That is, cf_LogRV_BetaNC evaluates the characteristic function cf(t) of Y = coef_i*log(X_1) +...+ coef_N*log(X_N), where X_i ~ Beta(\alpha_i,\beta_i,\delta_i) are inedependent RVs, with the shape parameters \alpha_i > 0, \beta_i > 0, and the noncentrality parameters \delta_i > 0, for i = 1,...,N.

For Type I noncentral beta distribution, the characteristic function of Y = log(X) with X ~ Beta(\alpha, \beta, \delta) is Poisson mixture of CFs of the central log-transformed Beta RVs of the form

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

where cf_LogRV_Beta(\alpha+j, \beta) are the CFs of central log-transformed Beta RVs with parameters \alpha+j and \beta. Alternatively,

cf(t) = cf_LogRV_BetaNC(t, \alpha, \beta, \delta) = Gamma(\alpha+1i*t)/Gamma(\alpha) * Gamma(\alpha+\beta)/Gamma(\alpha+\beta+1i*t) * exp(-\delta/2) * 2F2(\alpha+\beta, \alpha+1i*t; \alpha, \alpha+\beta+1i*t; \delta/2)

, where 2F2(a,b,;c,d;z) is hypergeometric function.

For Type II noncentral beta distribution, the characteristic function of Y = log(X) with X ~ Beta(\alpha, \beta, \delta) is Poisson mixture of CFs of the central log-transformed Beta RVs of the form

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

where cf_LogRV_Beta(\alpha, \beta+j) are the CFs of central log-transformed Beta RVs with parameters \alpha and \beta+j. Alternatively,

cf(t) = cf_LogRV_BetaNC(t, \alpha, \beta, \delta) = Gamma(\alpha+1i*t)/Gamma(\alpha) * Gamma(\alpha+\beta)/Gamma(\alpha+ \beta+1i*t) * 1F1(1i*t;1i*t+(\alpha+\beta); -\delta/2) = cf_LogRV_Beta(t, \alpha, \beta) * 1F1(1i*t;1i*t+(\alpha+\beta);-\delta/2)

where 1F1(a;b;z) is hypergeometric function.

Hence,the characteristic function of Y = coef(1)*Y1 + ... + coef(N)*YN is cf_Y(t) = cf_Y1(coef(1)*t) * ... * cf_YN(coef(N)*t), where cf_Yi(t) is evaluated with the parameters \alpha(i) and \beta(i).

Usage

cf_LogRV_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 LOG-TRANSFORMED non-central BETA random variables.

Note

Ver.: 20-Sep-2018 00:17:22 (consistent with Matlab CharFunTool v1.3.0, 06-Jul-2018 15:10:17).

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_BetaNC(), cf_BetaSymmetric(), cf_Beta(), cf_ChiSquare(), cf_Exponential(), cf_FisherSnedecorNC(), cf_FisherSnedecor(), cf_Gamma(), cf_InverseGamma(), cf_Laplace(), 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_BetaNC(), cf_FisherSnedecorNC(), cf_LogRV_ChiSquareNC(), cf_LogRV_FisherSnedecorNC()

Examples

## EXAMPLE 1
# CF of the log-transformed non-central Beta RV with delta = 1, coef = -1
alpha <- 1
beta  <- 3
delta <- 1
coef  <- -1
t <- seq(from = -10,
         to = 10,
         length.out = 201)
functions_to_plot <- list(function(t) cf_LogRV_BetaNC(t, alpha, beta, delta, coef, type = 1),
                          function(t) cf_LogRV_BetaNC(t, alpha, beta, delta, coef, type = 2))
plotReIm2(functions_to_plot, list(t, t),
          title = 'CF of minus log-transformed Type I and II Beta RV')


type <- 1
plotReIm(function(t)
        cf_LogRV_BetaNC(t, alpha, beta, delta, coef, type),
        t,
        title = 'CF of minus log-transformed Type I and II Beta RV')
par(new=TRUE)
type <- 2
plotReIm(function(t)
        cf_LogRV_BetaNC(t, alpha, beta, delta, coef, type),
        t,
        title = 'CF of minus log-transformed Type I and II Beta RV')

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

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

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