| loglaplace {VaRES} | R Documentation |
Log Laplace distribution
Description
Computes the pdf, cdf, value at risk and expected shortfall for the log Laplace distribution given by
\begin{array}{ll}
&\displaystyle
f (x) = \left\{
\begin{array}{ll}
\displaystyle
\frac {a b x^{b - 1}}{\delta^b (a + b)}, & \mbox{if $x \leq \delta$,}
\\
\\
\displaystyle
\frac {a b \delta^a}{x^{a + 1} (a + b)}, & \mbox{if $x > \delta$,}
\end{array}
\right.
\\
&\displaystyle
F (x) = \left\{
\begin{array}{ll}
\displaystyle
\frac {a x^b}{\delta^b (a + b)}, & \mbox{if $x \leq \delta$,}
\\
\\
\displaystyle
1 - \frac {b \delta^a}{x^a (a + b)}, & \mbox{if $x > \delta$,}
\end{array}
\right.
\\
&\displaystyle
{\rm VaR}_p (X) = \left\{
\begin{array}{ll}
\displaystyle
\delta \left[ p \frac {a + b}{a} \right]^{1/b}, & \mbox{if $p \leq \frac {a}{a + b}$,}
\\
\\
\displaystyle
\delta \left[ (1 - p) \frac {a + b}{a} \right]^{-1/a}, & \mbox{if $p > \frac {a}{a + b}$,}
\end{array}
\right.
\\
&\displaystyle
{\rm ES}_p (X) =
\left\{
\begin{array}{ll}
\displaystyle
\frac {\delta b}{b + 1}
\left[ p \frac {a + b}{a} \right]^{1/b}, & \mbox{if $p \leq \frac {a}{a + b}$,}
\\
\\
\displaystyle
\frac {a \delta}{p (1 + 1/b) (a + b)} +
\frac {a^{1/a} b^{1 - 1/a} \delta}{p (a + b) (1 - 1/a)}
\\
\displaystyle
\quad
-\frac {\delta (1 - p)}{p (1 - 1/a)}
\left[ \frac {a}{(a + b) (1 - p)} \right]^{1/a}, &
\mbox{if $p > \frac {a}{a + b}$}
\end{array}
\right.
\end{array}
for -\infty < x < \infty, 0 < p < 1, \delta > 0, the scale parameter,
a > 0, the first shape parameter, and b > 0, the second shape parameter.
Usage
dloglaplace(x, a=1, b=1, delta=0, log=FALSE)
ploglaplace(x, a=1, b=1, delta=0, log.p=FALSE, lower.tail=TRUE)
varloglaplace(p, a=1, b=1, delta=0, log.p=FALSE, lower.tail=TRUE)
esloglaplace(p, a=1, b=1, delta=0)
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 |
delta |
the value of the scale parameter, must be positive, the default is 1 |
a |
the value of the first shape parameter, must be positive, the default is 1 |
b |
the value of the second shape 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
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, doi:10.1080/03610918.2014.944658
Examples
x=runif(10,min=0,max=1)
dloglaplace(x)
ploglaplace(x)
varloglaplace(x)
esloglaplace(x)