cf_Student: 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 STUDENT's t random variables

Description

cf_Student(t, df, mu, sigma, coef, niid) evaluates the characteristic function cf(t) of a linear combination (resp. convolution) of independent (location and scale shifted) STUDENT's t random variables.

That is, cf_Student evaluates the characteristic function cf(t) of Y = sum_{i=1}^N coef_i * (mu_i + sigma_i * X_i), where X_i ~ t(df_i) are inedependent (symmetric) t-distributed RVs, with df_i > 0 degrees of freedom, for i = 1,...,N.

The characteristic function of the random variable \mu + \sigma*X, where X ~ t(df) is given by

cf(t) = exp(1i*t*mu) * besselk(df/2,abs(sigma*t)*sqrt(df),1) * exp(-abs(sigma*t)*sqrt(df)) * (sqrt(df)*abs(aigma*t))^(df/2) / 2^(df/2-1)/gamma(df/2).

Hence, the characteristic function of Y = coef_1*(\mu_1+\sigma_1*X_1) +...+ coef_N*(\mu_N+\sigma_N*X_N) is cf_Y(t) = exp(1i*\mu*t) * (cf_1(coef_1*\sigma_1*t) *...* cf_N(coef_N*\sigma_N*t)), where cf_i(t) is the characteristic function of X_i ~ t(df_i).

Usage

cf_Student(t, df, mu, sigma, coef, niid)

Arguments

t	vector or array of real values, where the CF is evaluated.
df	the degrees of freedom, df > 0. If empty, the default value is df = 1.
mu	vector of location parameters, mu in Real. If empty, default value is mu = 0.
sigma	vector of scale parameters, sigma_i > 0. If empty, default value is sigma = 1.
coef	vector of the coefficients of the linear combination of the STUDENT's t 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, such that Z = Y + ... + Y is sum of niid random variables Y, where each Y = sum_{i=1}^N coef_i * X_i is independently and identically distributed random variable. If empty, default value is niid = 1.

Value

Characteristic function cf(t) of a linear combination of independent STUDENT's t random variables.

Note

Ver.: 16-Sep-2018 18:38:54 (consistent with Matlab CharFunTool v1.3.0, 02-Jun-2017 12:08:24).

References

WITKOVSKY V. (2016). Numerical inversion of a characteristic function: An alternative tool to form the probability distribution of output quantity in linear measurement models. Acta IMEKO, 5(3), 32-44.

See Also

For more details see WIKIPEDIA: https://en.wikipedia.org/wiki/Student's_t-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_FisherSnedecorNC(), cf_LogRV_FisherSnedecor(), cf_LogRV_MeansRatioW(), cf_LogRV_MeansRatio(), cf_LogRV_WilksLambdaNC(), cf_LogRV_WilksLambda(), cf_Normal(), cf_RectangularSymmetric(), cf_TSPSymmetric(), cf_TrapezoidalSymmetric(), cf_TriangularSymmetric(), cf_vonMises()

Examples

## EXAMPLE 1
# CF of a linear combination of independent Student's t RVs
coef <- 1 / (1:50)
df   <- seq(50, 1, -1)
t <- seq(from = -1,
         to = 1,
         length.out = 201)
plotReIm(function(t)
        cf_Student(t, df, coef),
        t,
        title = "Characteristic function of the linear combination of t RVs")

## EXAMPLE 2
# CDF/PDF of a linear combination of independent Student's t RVs
coef <- 1 / (1:50)
df <- seq(50, 1, -1)
cf <- function(t)
        cf_Student(t, df, coef)
x <- seq(from = -50,
         to = 50,
         length.out = 100)
prob <- c(0.9, 0.95, 0.975, 0.99)
options <- list()
options$N <- 2 ^ 12
result <- cf2DistGP(cf, x, prob, options)
result

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