lmomco (version 2.4.14)

pdfln3: 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)

Value

Probability density (\(f\)) for \(x\).

Arguments

x

A real value vector.

para

The parameters from parln3 or vec2par.

Author

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)