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 (ξ, α, ν) computed by parst3. The probability density function is

f(x) = \frac{Γ(\frac{1}{2} + \frac{1}{2}ν)}{αν^{1/2}\,Γ(\frac{1}{2})Γ(\frac{1}{2}ν)}(1+t^2/ν)^{-(ν+1)/2}\mbox{,}

where f(x) is the probability density for quantile x, ξ is a location parameter, α is a scale parameter, and ν 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 ν ≥ 1000, one can use dnorm(X, mean=U, sd=A) and for U = ξ and A=α for 1.000001 ≤ ν ≤ 1000, one can use dt((X-U)/A, N)/A for N=ν. 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

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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

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## 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)

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