cf_LogRV_FisherSnedecorNC: Characteristic function of a linear combinationof independent...

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

Characteristic function of a linear combinationof independent LOG-TRANSFORMED non-central Fisher-Snedecor random variables

Description

cf_LogRV_FisherSnedecorNC(t, df1, df2, delta, coef, niid, tol) evaluates characteristic function of a linear combination (resp. convolution) of independent LOG-TRANSFORMED non-central Fisher-Snedecor random variables, with distributions F(df1_i,df2_i,\delta_i).

That is, cf_LogRV_FisherSnedecorNC evaluates the characteristic function cf(t) of Y = coef_i*log(X_1) +...+ coef_N*log(X_N), where X_i ~ F(df1_i,df2_i,\delta_i) are inedependent RVs, with df1_i and df2_i degrees of freedom, and the noncentrality parameters delta_i >0, for i = 1,...,N.

The characteristic function of Y = log(X) with X ~ F(df1,df2,\delta) is Poisson mixture of the CFs of the shifted log-transformed central F RVs of the form

cf(t) = cf_LogRV_FisherSnedecorNC(t,df1,df2,\delta) = exp(-\delta/2) sum_{j=1}^Inf (\delta/2)^j/j! * exp(1i*t*(df1+2*j)/df1) * cf_LogRV_FisherSnedecor(t,df1+2*j,df2),

where cf_LogRV_FisherSnedecor(t,df1,df2) denotes CF of log-transformed centrally distributed F RVs with parameters df1 and df2. For more details on the non-central F distribution see cf_FisherSnedecorNC. Alternatively,

cf(t) = (df2/df1)^(1i*t) * gamma(df1/2 + 1i*t) / gamma(df1/2) * gamma(df2/2 - 1i*t) / gamma(df2/2) * 1F1(-1i*t;df1/2;-delta/2),

where 1F1(a;b;z) is the confluent hypergeometric function, also known as the Kummer function M(a,b,z). 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 df1_i, df2_i, and delta_i.

Usage

cf_LogRV_FisherSnedecorNC(t, df1, df2, delta, coef, niid, tol)

Arguments

t	vector or array of real values, where the CF is evaluated.
df1	vector of the degrees of freedom df1 > 0. If empty, default value is df1 = 1.
df2	vector of the degrees of freedom df2 > 0. If empty, default value is df2 = 1.
delta	vector of non-centrality parameters.
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 Fisher-Snedecor random variables.

Note

Ver.: 20-Sep-2018 19:44:50 (consistent with Matlab CharFunTool v1.3.0, 10-Aug-2018 15:46:49).

See Also

For more details see WIKIPEDIA: https://en.wikipedia.org/wiki/Noncentral_F-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_ChiSquareNC(), cf_LogRV_ChiSquare(), 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_ChiSquareNC()

Examples

## EXAMPLE 1
# CF of the log-transformed non-central F RV with delta = 1 and coef = -1
df1 <- 3
df2 <- 5
delta <- 1
coef  <- -1
t <- seq(from = -10,
         to = 10,
         length.out = 201)
plotReIm(function(t)
        cf_LogRV_FisherSnedecorNC(t, df1, df2, delta, coef),
        t,
        title = 'CF of minus log-transformed F RV')

## EXAMPLE 2
# CDF/PDF of the minus log-transformed non-central F RV with delta = 1
df1 <- 3
df2 <- 5
delta <- 1
coef <- -1
cf <-
        function(t)
                cf_LogRV_FisherSnedecorNC(t, df1, df2, delta, coef)
options <- list()
options$N  <- 2 ^ 12
result <- cf2DistGP(cf = cf, options = options)

## EXAMPLE 3
# CDF/PDF of the linear combination of log-transformed non-central F RVs
df1 <- c(5, 4, 3)
df2 <- c(3, 4, 5)
delta <- c(0, 1, 2)
coef  <- -1 / 3
cf <-
        function(t)
                cf_LogRV_FisherSnedecorNC(t, df1, df2, delta, coef)
options <- list()
options$N  <- 2 ^ 12
result <- cf2DistGP(cf = cf, options = options)

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