pdfln3: Probability Density Function of the 3-Parameter Log-Normal...

pdfln3R Documentation

Probability Density Function of the 3-Parameter Log-Normal Distribution


This function computes the probability density of the Log-Normal3 distribution given parameters (ζ, lower bounds; μ_{\mathrm{log}}, location; and σ_{\mathrm{log}}, scale) computed by parln3. The probability density function function (same as Generalized Normal distribution, pdfgno) is

f(x) = \frac{\exp(κ Y - Y^2/2)}{α √{2π}} \mbox{,}

where Y is

Y = \frac{\log(x - ζ) - μ_{\mathrm{log}}}{σ_{\mathrm{log}}}\mbox{,}

where ζ is the lower bounds (real space) for which ζ < λ_1 - λ_2 (checked in are.parln3.valid), μ_{\mathrm{log}} be the mean in natural logarithmic space, and σ_{\mathrm{log}} be the standard deviation in natural logarithm space for which σ_{\mathrm{log}} > 0 (checked in are.parln3.valid) is obvious because this parameter has an analogy to the second product moment. Letting η = \exp(μ_{\mathrm{log}}), the parameters of the Generalized Normal are ζ + η, α = ησ_{\mathrm{log}}, and κ = -σ_{\mathrm{log}}. At this point, the algorithms (pdfgno) for the Generalized Normal provide the functional core.


pdfln3(x, para)



A real value vector.


The parameters from parln3 or vec2par.


Probability density (f) for x.


The parameterization of the Log-Normal3 results in ready support for either a known or unknown lower bounds. Details regarding the parameter fitting and control of the ζ parameter can be seen under the Details section in parln3.


W.H. Asquith


Asquith, W.H., 2011, Distributional analysis with L-moment statistics using the R environment for statistical computing: Createspace Independent Publishing Platform, ISBN 978–146350841–8.

See Also

cdfln3, qualn3, lmomln3, parln3, pdfgno


  lmr <- lmoms(c(123,34,4,654,37,78))
  ln3 <- parln3(lmr); gno <- pargno(lmr)
  x <- qualn3(0.5,ln3)
  pdfln3(x,ln3) # 0.008053616
  pdfgno(x,gno) # 0.008053616 (the distributions are the same, but see Note)

lmomco documentation built on Aug. 27, 2022, 1:06 a.m.