# Generalized asymmetric Student's t distribution 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 [REMOVE_ME]$$ \begin{array}{ll}&\displaystyle\displaystylef (x) = \left\{\begin{array}{ll}\displaystyle\frac {\alpha}{\alpha^{*}} K \left( \nu_1 \right) \left[ 1 + \frac {1}{\nu_1}\left( \frac {x}{2 \alpha^{*}} \right)^2 \right]^{-\frac {\nu_1 + 1}{2}}, & \mbox{if $x \leq 0$,}\\\\\displaystyle\frac {1 - \alpha}{1 - \alpha^{*}} K \left( \nu_2 \right)\left[ 1 + \frac {1}{\nu_2} \left( \frac {x}{2 \left( 1 - \alpha^{*} \right)} \right)^2 \right]^{-\frac {\nu_2 + 1}{2}}, &\mbox{if $x > 0$,}\end{array}\right.\\&\displaystyle\displaystyleF (x) = 2 \alpha F_{\nu_1} \left( \frac {\min (x, 0)}{2 \alpha^{*}} \right) -1 + \alpha + 2 (1 - \alpha)F_{\nu_2} \left( \frac {\max (x, 0)}{2 - 2 \alpha^{*}} \right),\\&\displaystyle\displaystyle{\rm VaR}_p (X) = 2 \alpha^{*} F_{\nu_1}^{-1} \left( \frac {\min (p, \alpha)}{2 \alpha} \right) +2 \left( 1 - \alpha^{*} \right) F_{\nu_2}^{-1} \left( \frac {\max (p, \alpha) + 1 - 2 \alpha}{2 - 2 \alpha} \right),\\&\displaystyle\displaystyle{\rm ES}_p (X) = \frac {2 \alpha^{*}}{p} \int_0^p F_{\nu_1}^{-1}\left( \frac {\min (v, \alpha)}{2 \alpha} \right) dv +\frac {2 \left( 1 - \alpha^{*} \right)}{p} \int_0^p F_{\nu_2}^{-1}\left( \frac {\max (v, \alpha) + 1 - 2 \alpha}{2 - 2 \alpha} \right) dv\end{array} [REMOVE_ME_2]$$ 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 $F_\nu(\cdot)$. ## Description 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 $$ \begin{array}{ll}&\displaystyle\displaystylef (x) = \left\{\begin{array}{ll}\displaystyle\frac {\alpha}{\alpha^{*}} K \left( \nu_1 \right) \left[ 1 + \frac {1}{\nu_1}\left( \frac {x}{2 \alpha^{*}} \right)^2 \right]^{-\frac {\nu_1 + 1}{2}}, & \mbox{if $x \leq 0$,}\\\\\displaystyle\frac {1 - \alpha}{1 - \alpha^{*}} K \left( \nu_2 \right)\left[ 1 + \frac {1}{\nu_2} \left( \frac {x}{2 \left( 1 - \alpha^{*} \right)} \right)^2 \right]^{-\frac {\nu_2 + 1}{2}}, &\mbox{if $x > 0$,}\end{array}\right.\\&\displaystyle\displaystyleF (x) = 2 \alpha F_{\nu_1} \left( \frac {\min (x, 0)}{2 \alpha^{*}} \right) -1 + \alpha + 2 (1 - \alpha)F_{\nu_2} \left( \frac {\max (x, 0)}{2 - 2 \alpha^{*}} \right),\\&\displaystyle\displaystyle{\rm VaR}_p (X) = 2 \alpha^{*} F_{\nu_1}^{-1} \left( \frac {\min (p, \alpha)}{2 \alpha} \right) +2 \left( 1 - \alpha^{*} \right) F_{\nu_2}^{-1} \left( \frac {\max (p, \alpha) + 1 - 2 \alpha}{2 - 2 \alpha} \right),\\&\displaystyle\displaystyle{\rm ES}_p (X) = \frac {2 \alpha^{*}}{p} \int_0^p F_{\nu_1}^{-1}\left( \frac {\min (v, \alpha)}{2 \alpha} \right) dv +\frac {2 \left( 1 - \alpha^{*} \right)}{p} \int_0^p F_{\nu_2}^{-1}\left( \frac {\max (v, \alpha) + 1 - 2 \alpha}{2 - 2 \alpha} \right) dv\end{array} $$ 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 $F_\nu(\cdot)$. ```r 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) ``` ## 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 - `alpha`: the value of the scale parameter, must be in the unit interval, the default is 0.5 - `nu1`: the value of the first degree of freedom parameter, must be positive, the default is 1 - `nu2`: the value of the second 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 ```r x=runif(10,min=0,max=1) dast(x) past(x) varast(x) esast(x) ```

ast function