Student's t distribution

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Description

Computes the pdf, cdf, value at risk and expected shortfall for the Student's t distribution due to Gosset (1908) given by

\begin{array}{ll} &\displaystyle f (x) = \frac {Γ ≤ft( \frac {n + 1}{2} \right)}{√{n π} Γ ≤ft( \frac {n}{2} \right)} ≤ft( 1 + \frac {x^2}{n} \right)^{-\frac {n + 1}{2}}, \\ &\displaystyle F (x) = \frac {1 + {\rm sign} (x)}{2} - \frac {{\rm sign} (x)}{2} I_{\frac {n}{x^2 + n}} ≤ft( \frac {n}{2}, \frac {1}{2} \right), \\ &\displaystyle {\rm VaR}_p (X) = √{n} {\rm sign} ≤ft( p - \frac {1}{2} \right) √{\frac {1}{I_a^{-1} ≤ft( \frac {n}{2}, \frac {1}{2} \right)} - 1}, \\ &\displaystyle \quad \mbox{ where $a = 2p$ if $p < 1/2$, $a = 2(1 - p)$ if $p ≥q 1/2$,} \\ &\displaystyle {\rm ES}_p (X) = \frac {√{n}}{p} \int_0^p {\rm sign} ≤ft( v - \frac {1}{2} \right) √{\frac {1}{I_a^{-1} ≤ft( \frac {n}{2}, \frac {1}{2} \right)} - 1} dv, \\ &\displaystyle \quad \mbox{ where $a = 2v$ if $v < 1/2$, $a = 2(1 - v)$ if $v ≥q 1/2$} \end{array}

for -∞ < x < ∞, 0 < p < 1, and n > 0, the degree of freedom parameter.

Usage

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dT(x, n=1, log=FALSE)
pT(x, n=1, log.p=FALSE, lower.tail=TRUE)
varT(p, n=1, log.p=FALSE, lower.tail=TRUE)
esT(p, n=1)

Arguments

x

scaler or vector of values at which the pdf or cdf needs to be computed

p

scaler or vector of values at which the value at risk or expected shortfall needs to be computed

n

the value of the degree of freedom parameter, must be positive, the default is 1

log

if TRUE then log(pdf) are returned

log.p

if TRUE then log(cdf) are returned and quantiles are computed for exp(p)

lower.tail

if FALSE then 1-cdf are returned and quantiles are computed for 1-p

Value

An object of the same length as x, giving the pdf or cdf values computed at x or an object of the same length as p, giving the values at risk or expected shortfall computed at p.

Author(s)

Saralees Nadarajah

References

S. Nadarajah, S. Chan and E. Afuecheta, An R Package for value at risk and expected shortfall, submitted

Examples

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x=runif(10,min=0,max=1)
dT(x)
pT(x)
varT(x)
esT(x)

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