lmomco (version 2.3.7)

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 $$f(x)=\frac{\mathrm{\Gamma }(\frac{1}{2}+\frac{1}{2}\nu )}{\alpha {\nu }^{1/2}\mathrm{\Gamma }(\frac{1}{2})\mathrm{\Gamma }(\frac{1}{2}\nu )}(1+t^2/\nu {)}^{-(\nu +1)/2}\text{,}$$ where $f(x)$ is the probability density for quantile $x$, $\xi$ 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)
}