pdfln3 {lmomco}

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

Description

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

f(x) = \frac{\exp(\kappa Y - Y^2/2)}{\alpha \sqrt{2\pi}} \mbox{,}

where Y is

Y = \frac{\log(x - \zeta) - \mu_{\mathrm{log}}}{\sigma_{\mathrm{log}}}\mbox{,}

where \zeta is the lower bounds (real space) for which \zeta < \lambda_1 - \lambda_2 (checked in are.parln3.valid), \mu_{\mathrm{log}} be the mean in natural logarithmic space, and \sigma_{\mathrm{log}} be the standard deviation in natural logarithm space for which \sigma_{\mathrm{log}} > 0 (checked in are.parln3.valid) is obvious because this parameter has an analogy to the second product moment. Letting \eta = \exp(\mu_{\mathrm{log}}), the parameters of the Generalized Normal are \zeta + \eta, \alpha = \eta\sigma_{\mathrm{log}}, and \kappa = -\sigma_{\mathrm{log}}. At this point, the algorithms (pdfgno) for the Generalized Normal provide the functional core.

Usage

pdfln3(x, para)

Arguments

x

A real value vector.

para

The parameters from parln3 or vec2par.

Value

Probability density (f) for x.

Note

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 \zeta parameter can be seen under the Details section in parln3.

Author(s)

W.H. Asquith

References

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

Examples

  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)