cf_FisherSnedecor: 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 random variables with the central FISHER-SNEDECOR F-distribution

Description

cf_FisherSnedecor(t, df1, df2, coef, niid, tol) evaluates characteristic function of the distribution of a linear combination of independent random variables with the central FISHER-SNEDECOR F-distribution.

That is, cf_FisherSnedecor evaluates the characteristic function cf(t) of Y = sum_{i=1}^N coef_i * X_i, where X_i ~ F(df1_i,df2_i) are inedependent RVs, with df1_i and df2_i degrees of freedom, for i = 1,...,N.

The characteristic function of X ~ F(df1,df2) is

cf(t) = U(df1/2, 1-df2/2, -1i*(df2/df1)*t),

where U(a,b,z) denotes the confluent hypergeometric function of the second kind.

Hence, the characteristic function of Y = coef_1*X_1 +...+ coef_N*X_N is

cf_Y(t) = cf_1(coef_1*t) *...* cf_N(coef_N*t),

where cf_i(t) is the characteristic function of X_i ~ F(df1_i,df2_i).

Usage

cf_FisherSnedecor(t, df1, df2, 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.
coef	vector of the coefficients of the linear combination of the log-transformed 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 random variables with the central FISHER-SNEDECOR F-distribution.

Note

Ver.: 16-Sep-2018 18:18:47 (consistent with Matlab CharFunTool v1.3.0, 24-Jun-2017 10:07:43).

References

[1] PHILLIPS, P.C.B. The true characteristic function of the F distribution. Biometrika (1982), 261-264.

[2] WITKOVSKY, V.: On the exact computation of the density and of the quantiles of linear combinations of t and F random variables. Journal of Statistical Planning and Inference 94 (2001), 1-13.

See Also

For more details see WIKIPEDIA: https://en.wikipedia.org/wiki/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_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()

Examples

## EXAMPLE 1
# CF of a linear combination of independent F RVs
df1 <- 3:12
df2 <- seq(14, 5,-1)
coef <- 1 / 10
t <- seq(from = -5,
         to = 5,
         length.out = 501)
plotReIm(function(t)
        cf_FisherSnedecor(t, df1, df2, coef),
        t,
        title = 'Characteristic function of the linear combination of F RVs')

## EXAMPLE 2
# PDF/CDF  of a linear combination of independent F RVs
df1 <- 3:12
df2 <- seq(14, 5,-1)
coef <- 1 / 10
cf <- function(t)
        cf_FisherSnedecor(t, df1, df2, coef)
options <- list()
options$N <- 2 ^ 10
options$xMin = 0
prob <- c(0.9, 0.95, 0.99)
result <- cf2DistGP(cf = cf, prob = prob, options = options)
result

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