Computes the pdf, cdf, value at risk and expected shortfall for the generalized beta distribution given by
\begin{array}{ll} &\displaystyle f (x) = \frac {(x - c)^{a - 1} (d - x)^{b - 1}}{B (a, b) (d - c)^{a + b - 1}}, \\ &\displaystyle F (x) = I_{\frac {x - c}{d - c}} (a, b), \\ &\displaystyle {\rm VaR}_p (X) = c + (d - c) I_p^{-1} (a, b), \\ &\displaystyle {\rm ES}_p (X) = c + \frac {d - c}{p} \int_0^p I_v^{-1} (a, b) dv \end{array}
for c ≤q x ≤q d, 0 < p < 1, a > 0, the first shape parameter, b > 0, the second shape parameter, -∞ < c < ∞, the first location parameter, and -∞ < c < d < ∞, the second location parameter.
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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 |
c |
the value of the first location parameter, can take any real value, the default is zero |
d |
the value of the second location parameter, can take any real value but must be greater than c, 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 |
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
S. Nadarajah, S. Chan and E. Afuecheta, An R Package for value at risk and expected shortfall, submitted
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