HBlaplace function

Holla-Bhattacharya Laplace distribution

Computes the pdf, cdf, value at risk and expected shortfall for the Holla-Bhattacharya Laplace distribution due to Holla and Bhattacharya (1968) given by [REMOVE_ME] \begin{array}{ll}&f (x) = \left\{ \begin{array}{ll}\displaystylea \phi \exp \left\{ \phi \left( x - \theta \right) \right\}, & \mbox{if $x \leq \theta$,}\\\\\displaystyle\left( 1 - a \right) \phi \exp \left\{ \phi \left( \theta - x \right) \right\}, & \mbox{if $x > \theta$,}\end{array}\right.\\&F (x) = \left\{\begin{array}{ll}\displaystylea \exp \left( \phi x - \theta \phi \right), & \mbox{if $x \leq \theta$,}\\\\\displaystyle1 - (1 - a) \exp \left( \theta \phi - \phi x \right), & \mbox{if $x > \theta$,}\end{array}\right.\\&{\rm VaR}_p (X) =\left\{\begin{array}{ll}\displaystyle\theta + \frac {1}{\phi} \log \left( \frac {p}{a} \right), &\mbox{if $p \leq a$,}\\\\\displaystyle\theta - \frac {1}{\phi} \log \left( \frac {1 - p}{1 - a} \right), &\mbox{if $p > a$,}\end{array}\right.\\&{\rm ES}_p (X) =\left\{\begin{array}{ll}\displaystyle\theta - \frac {1}{\phi} + \frac {1}{\phi} \log \frac {p}{a}, & \mbox{if $p \leq a$,}\\\\\displaystyle\frac {1}{p} \left[ \theta (1 + p - a) + \frac {p - 2a - (1 - a) \log a}{\phi} +\frac {1 - p}{\phi} \log \frac {1 - p}{1 - a} \right], & \mbox{if $p > a$}\end{array}\right.\end{array} [REMOVE_ME_2]

for $-\infty < x < \infty$, $0 < p < 1$, $-\infty < \theta < \infty$, the location parameter, $0 < a < 1$, the first scale parameter, and $\phi > 0$, the second scale parameter.

Description

Computes the pdf, cdf, value at risk and expected shortfall for the Holla-Bhattacharya Laplace distribution due to Holla and Bhattacharya (1968) given by

for $-\infty < x < \infty$, $0 < p < 1$, $-\infty < \theta < \infty$, the location parameter, $0 < a < 1$, the first scale parameter, and $\phi > 0$, the second scale parameter.

dHBlaplace(x, a=0.5, theta=0, phi=1, log=FALSE)
pHBlaplace(x, a=0.5, theta=0, phi=1, log.p=FALSE, lower.tail=TRUE)
varHBlaplace(p, a=0.5, theta=0, phi=1, log.p=FALSE, lower.tail=TRUE)
esHBlaplace(p, a=0.5, theta=0, phi=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
• theta: the value of the location parameter, can take any real value, the default is zero
• a: the value of the first scale parameter, must be in the unit interval, the default is 0.5
• phi: the value of the second scale 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)
dHBlaplace(x)
pHBlaplace(x)
varHBlaplace(x)
esHBlaplace(x)