cf_LogRV_ChiSquareNC: 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 ChiSquare random variables

Description

cf_LogRV_ChiSquareNC(t, df, delta, coef, niid, tol) evaluates characteristic function of a linear combination (resp. convolution) of independent LOG-TRANSFORMED non-central ChiSquare random variables, with distributions ChiSquare(df_i,\delta_i).

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

The characteristic function of Y = log(X) with X ~ ChiSquare(df,\delta) is Poisson mixture of CFs of the central log-transformed ChiSquare RVs of the form

cf(t) = cf_LogRV_ChiSquareNC(t,df,\delta) = exp(-\delta/2) sum_{j=1}^Inf (\delta/2)^j/j! * cf_LogRV_ChiSquare(df+2*j),

where cf_LogRV_ChiSquare(df+2+j) are the CFs of central log-transformed ChiSquare RVs with df+2+j degrees of freedom. Alternatively,

cf(t) = cf_LogRV_ChiSquareNC(t,df,\delta) = cf_LogRV_ChiSquare(t,df) * 1F1(-1i*t;df/2;-\delta/2),

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

Hence, the characteristic function of Y = coef(1)*Y1 + ... + coef(N)*YN

cf_Y(t) = cf_Y1(coef(1)*t) * ... * cf_YN(coef(N)*t),

where cf_Yi(t) is evaluated with the parameters df_i and delta_i.

Usage

cf_LogRV_ChiSquareNC(t, df, delta, coef, niid, tol)

Arguments

t	vector or array of real values, where the CF is evaluated.
df	vector of the degrees of freedom df > 0. If empty, default value is df = 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.

Value

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

Note

Ver.: 20-Sep-2018 19:53:05 (consistent with Matlab CharFunTool v1.3.0, 10-Aug-2018 22:25:01).

See Also

For more details see WIKIPEDIA: https://en.wikipedia.org/wiki/Noncentral_chi-squared_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_BetaNC(), cf_LogRV_Beta(), 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_BetaNC(), cf_LogRV_FisherSnedecorNC()

Examples

## EXAMPLE 1
# CF of the minus log-transformed non-central ChiSquare RV with delta = 1
df <- 3
delta <- 1
coef  <- -1
t <- seq(from = -10,
         to = 10,
         length.out = 201)
plotReIm(function(t)
        cf_LogRV_ChiSquareNC(t, df, delta, coef),
        t,
        title = 'CF of minus log-transformed ChiSquare RV')

## EXAMPLE 2
# CDF/PDF of the minus log-transformed non-central ChiSquare RV
df <- 3
delta <- 1
coef <- -1
cf <- function(t)
        cf_LogRV_ChiSquareNC(t, df, delta, coef)
options <- list()
options$N <- 2 ^ 10
result <- cf2DistGP(cf = cf, options = options)

## EXAMPLE 3
# CDF/PDF of the linear combination of the minus log-transformed non-central ChiSquare RVs
df <- c(3, 4, 5)
delta <- c(0, 1, 2)
coef  <- c(-1,-1,-1) / 3
cf <- function(t)
        cf_LogRV_ChiSquareNC(t, df, delta, coef)
options <- list()
options$N <- 2 ^ 10
result <- cf2DistGP(cf = cf, options = options)

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