lmomco (version 2.4.14)

pdfst3: 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 \(\nu \ge 1000\), one can use dnorm(X, mean=U, sd=A) and for U = \(\xi\) and A=\(\alpha\) for \(1.000001 \le \nu \le 1000\), one can use dt((X-U)/A, N)/A for N=\(\nu\). The R function dnorm is used for the Normal distribution and the R function dt is used for the 1-parameter Student t distribution.

Usage

pdfst3(x, para, paracheck=TRUE)

Value

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

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.

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

cdfst3, quast3, lmomst3, parst3

Examples

if (FALSE) {
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)
}