weibull | R Documentation |
Computes the pdf, cdf, value at risk and expected shortfall for the Weibull distribution due to Weibull (1951) given by
\begin{array}{ll}
&\displaystyle
f (x) = \frac {\alpha x^{\alpha - 1}}{\sigma^\alpha}
\exp \left\{ -\left( \frac {x}{\sigma} \right)^{\alpha} \right\},
\\
&\displaystyle
F (x) = 1 - \exp \left\{ -\left( \frac {x}{\sigma} \right)^{\alpha} \right\},
\\
&\displaystyle
{\rm VaR}_p (X) = \sigma \left[ -\log (1 - p) \right]^{1 / \alpha},
\\
&\displaystyle
{\rm ES}_p (X) = \frac {\sigma}{p} \gamma \left( 1 + 1 / \alpha, - \log (1 - p) \right)
\end{array}
for x > 0
, 0 < p < 1
, \alpha > 0
, the shape parameter, and \sigma > 0
, the scale parameter.
dWeibull(x, alpha=1, sigma=1, log=FALSE)
pWeibull(x, alpha=1, sigma=1, log.p=FALSE, lower.tail=TRUE)
varWeibull(p, alpha=1, sigma=1, log.p=FALSE, lower.tail=TRUE)
esWeibull(p, 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 |
alpha |
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)
dWeibull(x)
pWeibull(x)
varWeibull(x)
esWeibull(x)
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