cf_FisherSnedecorNC: 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 Fisher-Snedecor random variables

Description

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

That is, cf_FisherSnedecorNC evaluates the characteristic function cf(t) of Y = coef_i*X_1 +...+ coef_N*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.

Random variable X has non-central F distribution with df1 and df2 degrees of freedom and the non-centrality parameter \delta if X = (X1/df1)/(X2/df2) where X1 ~ ChiSquare(df1, \delta), i.e. with non-central chi-square distribution, and X2 ~ ChiSquare(df2) is independent random variable with central chi-square distribution.

The characteristic function of X ~ F(df1,df2,delta) is Poisson mixture of the CFs of the scaled central F RVs of the form

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

where cf_FisherSnedecor(t,df1,df2) are the CFs of central F RVs with parameters df1 and df2. 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_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	non-centrality parameter.
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 non-central Fisher-Snedecor random variables.

Note

Ver.: 20-Sep-2018 00:03:52 (consistent with Matlab CharFunTool v1.3.0, 10-Aug-2018 16:04:30).

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_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_BetaNC(), cf_LogRV_BetaNC(), cf_LogRV_ChiSquareNC(), cf_LogRV_FisherSnedecorNC()

Examples

## EXAMPLE 1
# CF of the non-central F RV with delta = 1
df1 <- 3
df2 <- 5
delta <- 1
t <- seq(from = -10,
         to = 10,
         length.out = 201)
plotReIm(function(t)
        cf_FisherSnedecorNC(t, df1, df2, delta), t, title = 'CF of non-central Fisher Snedecor RV')

## EXAMPLE 2
# CDF/PDF of the non-central F RV with delta = 1
df1 <- 3
df2 <- 5
delta <- 1
cf <- function(t)
        cf_FisherSnedecorNC(t, df1, df2, delta)
options <- list()
options$xMin <- 0
result <- cf2DistGP(cf = cf, options = options)

## EXAMPLE 3
# CDF/PDF of the linear combination of 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_FisherSnedecorNC(t, df1, df2, 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.