cdfln3: Cumulative Distribution Function of the 3-Parameter Log-Normal Distribution

Description

This function computes the cumulative probability or nonexceedance probability of the Log-Normal3 distribution given parameters ($\zeta$, lower bounds; $\mu_{\mathrm{log}}$, location; and $\sigma_{\mathrm{log}}$, scale) computed by parln3. The cumulative distribution function (same as Generalized Normal distribution, cdfgno) is $$F(x) = \Phi(Y) \mbox{,} $$ where $\Phi$ is the cumulative ditribution function of the Standard Normal distribution and $Y$ 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 (cdfgno) for the Generalized Normal provide the functional core.

Usage

cdfln3(x, para)

Value

Nonexceedance probability ($F$) 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.

Examples

Run this code

  lmr <- lmoms(c(123,34,4,654,37,78))
  cdfln3(50,parln3(lmr))

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