cfS_Student: Characteristic function of the STUDENT's t-distribution

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

Characteristic function of the STUDENT's t-distribution

Description

cfS_Student(t, df, mu, sigma, coef, niid) evaluates the characteristic function cf(t) of the STUDENT's t-distribution with df > 0 degrees of freedom.

cfS_Student is an ALIAS of the more general function cf_Student, used to evaluate the characteristic function of a linear combination of independent (location-scale) STUDENT's t-distributed random variables.

The characteristic function of the STUDENT's t-distribution with df degrees of freedom is defined by

cf(t) = cfS_Student(t,df) = besselk(df/2,abs(t)*sqrt(df),1) * exp(-abs(t)*sqrt(df)) * (sqrt(df)*abs(t))^(df/2) / 2^(df/2-1)/gamma(df/2).

Usage

cfS_Student(t, df = 1, 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.
sigma	vector.
coef	vector of coefficients of the linear combination of Student 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.

Value

Characteristic function cf(t) of the STUDENT's t-distribution.

Note

Ver.: 16-Sep-2018 19:09:41 (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_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_Student(), cf_TSPSymmetric(), cf_TrapezoidalSymmetric(), cf_TriangularSymmetric(), cf_vonMises()

Other Symmetric Probability Distribution: cfS_Arcsine(), cfS_Beta(), cfS_Gaussian(), cfS_Laplace(), cfS_Rectangular(), cfS_Trapezoidal(), cf_ArcsineSymmetric(), cf_BetaSymmetric(), cf_RectangularSymmetric(), cf_TSPSymmetric(), cf_TrapezoidalSymmetric()

Examples

## EXAMPLE1
# CF of the Student t-distribution with df = 3
df <- 3
t <- seq(-5, 5, length.out = 501)
plotReIm(function(t)
        cfS_Student(t, df), t, title = "CF of the Student t-distribution with df = 3")

## EXAMPLE2
# PDF/CDF of the Student t-distribution with df = 3
df <- 3
cf <- function(t)
        cfS_Student(t, df)
x <- seq(-8, 8, length.out = 101)
prob <- c(0.9, 0.95, 0.99)
options <- list()
options$N <- 2 ^ 12
options$SixSigmaRule <- 30
result <- cf2DistGP(cf, x, prob, options)

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