cfX_LogNormal: Characteristic function of Lognormal distribution

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

Characteristic function of Lognormal distribution

Description

cfX_LogNormal(t, mu, sigma, tol) computes the characteristic function cf(t) of the Lognormal distribution with parameters mu (real) and sigma > 0, computed for real (vector) argument t, i.e.

cf(t) = cfX_LogNormal(t,mu,sigma).

Usage

cfX_LogNormal(t, mu = 0, sigma = 1, tol = 1e-06)

Arguments

t	vector or array of real values, where the CF is evaluated.
mu	real, default value mu = 0.
sigma	> 0, default value sigma = 1.
tol	tolerance, default value tol = 1e-6.

Details

In probability theory, a log-normal (or lognormal) distribution is a continuous probability distribution of a random variable whose logarithm is normally distributed. The lognormal distribution is defined for x in (0,+inf) by its PDF/CDF/CF, as follows pdf(x) = 1/(x*\sigma*sqrt(2*pi))*exp(-(ln(x)-\mu)^2/(2*\sigma^2)) cdf(x) = 1/2+1/2*erf((ln(x)-\mu)/(sqrt(2)*\sigma)) cf(t) = sum_0^infinity{(it)^n/n!*exp(n*\mu + (n*\sigma)^2/2)}. As noted, this representation is asymptotically divergent but sufficient for numerical purposes.

cfX_LogNormal is based on the standard integral representation of the characteristic function of the lognormal distribution, i.e. cf(t) = Integral_0^inf exp(i*t*x) * PDF(x) dx. By using the half-space Fourier integral transformation we get cf(t) = Integral_0^inf (i/t) * exp(-x) * PDF(i*x/t) dx. If we define the integrand as funCF(t,x) = (i/t) * exp(-x) * PDF(i*x/t), then by using a stabilizing transformation from [0,inf] to [0,1], we can evaluate the CF by the following (well behaved) integral: cf(t) = Integral_0^1 2x/(1-x)^3 * funCF(t,(x/(1-x))^2) dx.

cfX_LogNormal evaluates this integral by using the R built in function integrate(), with precission specified by tolerance tol (default value is tol = 1e-6).

Value

Characteristic function cf(t) of the Lognormal distribution.

Note

Ver.: 16-Sep-2018 19:26:07 (consistent with Matlab CharFunTool v1.3.0, 15-Nov-2016 13:36:26).

References

[1] 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.

[2] 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.

[3] WITKOVSKY V., WIMMER G., DUBY T. (2016). Computing the aggregate loss distribution based on numerical inversion of the compound empirical characteristic function of frequency and severity. Working Paper. Insurance: Mathematics and Economics.

[4] DUBY T., WIMMER G., WITKOVSKY V.(2016). MATLAB toolbox CRM for computing distributions of collective risk models. Working Paper. Journal of Statistical Software.

See Also

For more details see WIKIPEDIA: https://en.wikipedia.org/wiki/Log-normal_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(), 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()

Examples

## EXAMPLE1
# CF of the Lognormal distribution with mu = 0,sigma = 1
mu <- 0
sigma <- 1
t <- seq(-20, 20, length.out = 2 ^ 10 + 1)
plotReIm(function(t)
        cfX_LogNormal(t, mu, sigma),
        t,
        title = "Characteristic function of the Lognormal distribution")

## EXAMPLE2
# CDF/PDF of the Lognormal distribution with mu = 0,sigma = 1
mu <- 0
sigma <- 1
x <- seq(0, 15, length.out = 101)
prob <- c(0.9, 0.95, 0.99)
cf <- function(t)
        cfX_LogNormal(t, mu, sigma)
options <- list()
options$xMin <- 0
options$N <- 2 ^ 13
options$SixSigmaRule <- 8
result <- cf2DistGP(cf, x, prob, options)

## EXAMPLE3
# PDF/CDF of the compound Poisson-Lognormal distribution
mu <- 0
sigma <- 1
lambda <- 10
x <- seq(0, 70, length.out = 101)
prob <- c(0.9, 0.95, 0.99)
cfX <- function(t)
        cfX_LogNormal(t, mu, sigma)
cf <- function(t)
        cfN_Poisson(t, lambda, cfX)
options <- list()
options$isCompound <- TRUE
result <- cf2DistGP(cf, x, prob, options)

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