pdfst3: Probability Density Function of the 3-Parameter Student t...

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Probability Density Function of the 3-Parameter Student t Distribution

Description

This function computes the probability density of the 3-parameter Student t distribution given parameters (\xi, \alpha, \nu) computed by parst3. The probability density function is

f(x) = \frac{\Gamma(\frac{1}{2} + \frac{1}{2}\nu)}{\alpha\nu^{1/2}\,\Gamma(\frac{1}{2})\Gamma(\frac{1}{2}\nu)}(1+t^2/\nu)^{-(\nu+1)/2}\mbox{,}

where f(x) is the probability density for quantile x, t is defined as t = (x - \xi)/\alpha, \xi is a location parameter, \alpha is a scale parameter, and \nu is a shape parameter in terms of the degrees of freedom as for the more familiar Student t distribution in R.

For value X, the built-in R functions can be used. For U = \xi and A=\alpha for 1.001 \le \nu \le 10^5.5, one can use dt((X-U)/A, N)/A for N=\nu. The R function dt is used for the 1-parameter Student t density. The limits for \nu stem from study of ability for theoretical integration of the quantile function to produce viable \tau_4 and \tau_6 (see inst/doc/t4t6/studyST3.R).

Usage

pdfst3(x, para, paracheck=TRUE)

Arguments

x	A real value vector.
para	The parameters from parst3 or vec2par.
paracheck	A logical on whether the parameter should be check for validity.

Value

Probability density (f) for x.

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

cdfst3, quast3, lmomst3, parst3

Examples

## Not run: 
xs <- -200:200
  para <- vec2par(c(37, 25,  114), type="st3")
plot(xs, pdfst3(xs, para), type="l")
  para <- vec2par(c(11, 36, 1000), type="st3")
lines(xs, pdfst3(xs, para), lty=2)
  para <- vec2par(c(-7, 60,   40), type="st3")
lines(xs, pdfst3(xs, para), lty=3)

## End(Not run)

lmomco documentation built on Nov. 7, 2025, 5:11 p.m.