expweibull | R Documentation |
Computes the pdf, cdf, value at risk and expected shortfall for the exponentiated Weibull distribution due to Mudholkar and Srivastava (1993) and Mudholkar et al. (1995) given by
\begin{array}{ll}
&\displaystyle
f(x) = a \alpha \sigma^{-\alpha} x^{\alpha - 1}
\exp \left[ -(x / \sigma)^\alpha \right]
\left\{ 1 - \exp \left[ -(x / \sigma)^\alpha \right] \right\}^{a - 1},
\\
&\displaystyle
F (x) = \left\{ 1 - \exp \left[ -(x / \sigma)^\alpha \right] \right\}^a,
\\
&\displaystyle
{\rm VaR}_p (X) = \sigma \left[ -\log \left( 1 - p^{1 / a} \right) \right]^{1 / \alpha},
\\
&\displaystyle
{\rm ES}_p (X) = \frac {\sigma}{p} \int_0^p \left[ -\log \left( 1 - v^{1 / a} \right) \right]^{1 / \alpha} dv
\end{array}
for x > 0
, 0 < p < 1
, a > 0
, the first shape parameter,
\alpha > 0
, the second shape parameter, and \sigma > 0
, the scale parameter.
dexpweibull(x, a=1, alpha=1, sigma=1, log=FALSE)
pexpweibull(x, a=1, alpha=1, sigma=1, log.p=FALSE, lower.tail=TRUE)
varexpweibull(p, a=1, alpha=1, sigma=1, log.p=FALSE, lower.tail=TRUE)
esexpweibull(p, a=1, alpha=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 |
sigma |
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 |
alpha |
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 |
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)
dexpweibull(x)
pexpweibull(x)
varexpweibull(x)
esexpweibull(x)
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