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

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Description

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

F(x) = Φ(Y) \mbox{,}

where Φ is the cumulative ditribution function of the Standard Normal distribution and 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 (cdfgno) for the Generalized Normal provide the functional core.

Usage

1
cdfln3(x, para)

Arguments

x

A real value vector.

para

The parameters from parln3 or vec2par.

Value

Nonexceedance probability (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 ζ 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

pdfln3, qualn3, lmomln3, parln3, cdfgno

Examples

1
2
  lmr <- lmoms(c(123,34,4,654,37,78))
  cdfln3(50,parln3(lmr))

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