T function

Student's t distribution

Computes the pdf, cdf, value at risk and expected shortfall for the Student's $t$ distribution due to Gosset (1908) given by [REMOVE_ME] \begin{array}{ll}&\displaystylef (x) = \frac {\Gamma \left( \frac {n + 1}{2} \right)}{\sqrt{n \pi} \Gamma \left( \frac {n}{2} \right)}\left( 1 + \frac {x^2}{n} \right)^{-\frac {n + 1}{2}},\\&\displaystyleF (x) = \frac {1 + {\rm sign} (x)}{2} - \frac {{\rm sign} (x)}{2} I_{\frac {n}{x^2 + n}} \left( \frac {n}{2}, \frac {1}{2} \right),\\&\displaystyle{\rm VaR}_p (X) = \sqrt{n} {\rm sign} \left( p - \frac {1}{2} \right)\sqrt{\frac {1}{I_a^{-1} \left( \frac {n}{2}, \frac {1}{2} \right)} - 1},\\&\displaystyle\quad\mbox{ where $a = 2p$ if $p < 1/2$, $a = 2(1 - p)$ if $p \geq 1/2$,}\\&\displaystyle{\rm ES}_p (X) = \frac {\sqrt{n}}{p} \int_0^p {\rm sign} \left( v - \frac {1}{2} \right)\sqrt{\frac {1}{I_a^{-1} \left( \frac {n}{2}, \frac {1}{2} \right)} - 1} dv,\\&\displaystyle\quad\mbox{ where $a = 2v$ if $v < 1/2$, $a = 2(1 - v)$ if $v \geq 1/2$}\end{array} [REMOVE_ME_2]

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

Description

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

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

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

Returns

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.

References

Stephen Chan, Saralees Nadarajah & Emmanuel Afuecheta (2016). An R Package for Value at Risk and Expected Shortfall, Communications in Statistics - Simulation and Computation, 45:9, 3416-3434, tools:::Rd_expr_doi("10.1080/03610918.2014.944658")

Author(s)

Saralees Nadarajah

Examples

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