CharFun (version 0.1.0)

cfX_LogNormal: Characteristic function of Lognormal distribution

Description

cfX_LogNormal(t,mu,sigma) 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

numerical values (number, vector...)

mu

real, default value mu = 0

sigma

> 0, default value sigma = 1

tol

tolerance, default value tol = 1e-6

Value

characteristic function cf(t) of the Lognormal distribution computed for real (vector)

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).

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_Rectangular, cfS_StudentT, cfS_Trapezoidal, cfS_Triangular, cfX_Beta, cfX_ChiSquared, cfX_Exponential, cfX_Gamma, cfX_InverseGamma, cfX_Normal, cfX_PearsonV, cfX_Rectangular, cfX_Triangular

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)
plotGraf(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)
result <-
  cf2DistGP(cf,
            x,
            prob,
            N = 2 ^ 10,
            SixSigmaRule = 8,
            xMin = 0)

# 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)
result <- cf2DistGP(cf, x, prob, isCompound = TRUE, N = 2 ^ 10)