loglaplace function

Log Laplace distribution

Computes the pdf, cdf, value at risk and expected shortfall for the log Laplace distribution given by [REMOVE_ME] \begin{array}{ll}&\displaystylef (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.\\&\displaystyleF (x) = \left\{\begin{array}{ll}\displaystyle\frac {a x^b}{\delta^b (a + b)}, & \mbox{if $x \leq \delta$,}\\\\\displaystyle1 - \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} [REMOVE_ME_2]

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.

Description

Computes the pdf, cdf, value at risk and expected shortfall for the log Laplace distribution given by

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.

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

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)
dloglaplace(x)
ploglaplace(x)
varloglaplace(x)
esloglaplace(x)