paretostable | R Documentation |
Computes the pdf, cdf, value at risk and expected shortfall for the Pareto positive stable distribution due to Sarabia and Prieto (2009) and Guillen et al. (2011) given by
\begin{array}{ll}
&\displaystyle
f (x) = \frac {\nu \lambda}{x}
\left[ \log \left( \frac {x}{\sigma} \right) \right]^{\nu - 1}
\exp \left\{ -\lambda \left[ \log \left( \frac {x}{\sigma} \right) \right]^\nu \right\},
\\
&\displaystyle
F (x) = 1 - \exp \left\{ -\lambda \left[ \log \left( \frac {x}{\sigma} \right) \right]^\nu \right\},
\\
&\displaystyle
{\rm VaR}_p (X) = \sigma \exp \left\{ \left[ -\frac {1}{\lambda} \log (1 - p) \right]^{1/\nu} \right\},
\\
&\displaystyle
{\rm ES}_p (X) = \frac {\sigma}{p} \int_0^p \exp \left\{ \left[ -\frac {1}{\lambda} \log (1 - v) \right]^{1/\nu} \right\} dv
\end{array}
for x > 0
, 0 < p < 1
, \lambda > 0
, the first scale parameter, \sigma > 0
, the second scale parameter, and \nu > 0
, the shape parameter.
dparetostable(x, lambda=1, nu=1, sigma=1, log=FALSE)
pparetostable(x, lambda=1, nu=1, sigma=1, log.p=FALSE, lower.tail=TRUE)
varparetostable(p, lambda=1, nu=1, sigma=1, log.p=FALSE, lower.tail=TRUE)
esparetostable(p, lambda=1, nu=1, sigma=1)
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 |
lambda |
the value of the first scale parameter, must be positive, the default is 1 |
sigma |
the value of the second scale parameter, must be positive, the default is 1 |
nu |
the value of the 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 |
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
.
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, \Sexpr[results=rd]{tools:::Rd_expr_doi("10.1080/03610918.2014.944658")}
x=runif(10,min=0,max=1)
dparetostable(x)
pparetostable(x)
varparetostable(x)
esparetostable(x)
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