Computes the pdf, cdf, value at risk and expected shortfall for the generalized asymmetric Student's \(t\) distribution due to Zhu and Galbraith (2010) given by $$\begin{array}{ll} &\displaystyle \displaystyle f (x) = \left\{ \begin{array}{ll} \displaystyle \frac {\alpha}{\alpha^{*}} K \left( \nu_1 \right) \left[ 1 + \frac {1}{\nu_1} \left( \frac {x}{2 \alpha^{*}} \right)^2 \right]^{-\frac {\nu_1 + 1}{2}}, & \mbox{if $x \leq 0$,} \\ \\ \displaystyle \frac {1 - \alpha}{1 - \alpha^{*}} K \left( \nu_2 \right) \left[ 1 + \frac {1}{\nu_2} \left( \frac {x}{2 \left( 1 - \alpha^{*} \right)} \right)^2 \right]^{-\frac {\nu_2 + 1}{2}}, & \mbox{if $x > 0$,} \end{array} \right. \\ &\displaystyle \displaystyle F (x) = 2 \alpha F_{\nu_1} \left( \frac {\min (x, 0)}{2 \alpha^{*}} \right) -1 + \alpha + 2 (1 - \alpha) F_{\nu_2} \left( \frac {\max (x, 0)}{2 - 2 \alpha^{*}} \right), \\ &\displaystyle \displaystyle {\rm VaR}_p (X) = 2 \alpha^{*} F_{\nu_1}^{-1} \left( \frac {\min (p, \alpha)}{2 \alpha} \right) + 2 \left( 1 - \alpha^{*} \right) F_{\nu_2}^{-1} \left( \frac {\max (p, \alpha) + 1 - 2 \alpha}{2 - 2 \alpha} \right), \\ &\displaystyle \displaystyle {\rm ES}_p (X) = \frac {2 \alpha^{*}}{p} \int_0^p F_{\nu_1}^{-1} \left( \frac {\min (v, \alpha)}{2 \alpha} \right) dv + \frac {2 \left( 1 - \alpha^{*} \right)}{p} \int_0^p F_{\nu_2}^{-1} \left( \frac {\max (v, \alpha) + 1 - 2 \alpha}{2 - 2 \alpha} \right) dv \end{array}$$ for \(-\infty < x < \infty\), \(0 < p < 1\), \(0 < \alpha < 1\), the scale parameter, \(\nu_1 > 0\), the first degree of freedom parameter, and \(\nu_2 > 0\), the second degree of freedom parameter, where \(\alpha^{*} = \alpha K \left( \nu_1 \right) / \left\{ \alpha K \left( \nu_1 \right) + (1 - \alpha) K \left( \nu_2 \right) \right\}\), \(K (\nu) = \Gamma \left( (\nu + 1)/2 \right) / \left[ \sqrt{\pi \nu} \Gamma (\nu/2) \right]\), \(F_\nu(\cdot)\) denotes the cdf of a Student's \(t\) random variable with \(\nu\) degrees of freedom, and \(F_\nu^{-1} (\cdot)\) denotes the inverse of \(F_\nu(\cdot)\).
dast(x, nu1=1, nu2=1, alpha=0.5, log=FALSE)
past(x, nu1=1, nu2=1, alpha=0.5, log.p=FALSE, lower.tail=TRUE)
varast(p, nu1=1, nu2=1, alpha=0.5, log.p=FALSE, lower.tail=TRUE)
esast(p, nu1=1, nu2=1, alpha=0.5)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.
scaler or vector of values at which the pdf or cdf needs to be computed
scaler or vector of values at which the value at risk or expected shortfall needs to be computed
the value of the scale parameter, must be in the unit interval, the default is 0.5
the value of the first degree of freedom parameter, must be positive, the default is 1
the value of the second degree of freedom parameter, must be positive, the default is 1
if TRUE then log(pdf) are returned
if TRUE then log(cdf) are returned and quantiles are computed for exp(p)
if FALSE then 1-cdf are returned and quantiles are computed for 1-p
Saralees Nadarajah
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")
x=runif(10,min=0,max=1)
dast(x)
past(x)
varast(x)
esast(x)Run the code above in your browser using DataLab