weibullgpd {evmix} | R Documentation |
Weibull Bulk and GPD Tail Extreme Value Mixture Model
Description
Density, cumulative distribution function, quantile function and
random number generation for the extreme value mixture model with Weibull for bulk
distribution upto the threshold and conditional GPD above threshold. The parameters
are the weibull shape wshape
and scale wscale
, threshold u
GPD scale sigmau
and shape xi
and tail fraction phiu
.
Usage
dweibullgpd(x, wshape = 1, wscale = 1, u = qweibull(0.9, wshape,
wscale), sigmau = sqrt(wscale^2 * gamma(1 + 2/wshape) - (wscale *
gamma(1 + 1/wshape))^2), xi = 0, phiu = TRUE, log = FALSE)
pweibullgpd(q, wshape = 1, wscale = 1, u = qweibull(0.9, wshape,
wscale), sigmau = sqrt(wscale^2 * gamma(1 + 2/wshape) - (wscale *
gamma(1 + 1/wshape))^2), xi = 0, phiu = TRUE, lower.tail = TRUE)
qweibullgpd(p, wshape = 1, wscale = 1, u = qweibull(0.9, wshape,
wscale), sigmau = sqrt(wscale^2 * gamma(1 + 2/wshape) - (wscale *
gamma(1 + 1/wshape))^2), xi = 0, phiu = TRUE, lower.tail = TRUE)
rweibullgpd(n = 1, wshape = 1, wscale = 1, u = qweibull(0.9,
wshape, wscale), sigmau = sqrt(wscale^2 * gamma(1 + 2/wshape) - (wscale
* gamma(1 + 1/wshape))^2), xi = 0, phiu = TRUE)
Arguments
x |
quantiles |
wshape |
Weibull shape (positive) |
wscale |
Weibull scale (positive) |
u |
threshold |
sigmau |
scale parameter (positive) |
xi |
shape parameter |
phiu |
probability of being above threshold |
log |
logical, if TRUE then log density |
q |
quantiles |
lower.tail |
logical, if FALSE then upper tail probabilities |
p |
cumulative probabilities |
n |
sample size (positive integer) |
Details
Extreme value mixture model combining Weibull distribution for the bulk below the threshold and GPD for upper tail.
The user can pre-specify phiu
permitting a parameterised value for the tail fraction . Alternatively, when
phiu=TRUE
the tail fraction is estimated as the tail fraction from the
weibull bulk model.
The cumulative distribution function with tail fraction defined by the
upper tail fraction of the Weibull bulk model (
phiu=TRUE
), upto the
threshold , given by:
and above the threshold :
where and
are the Weibull and conditional GPD
cumulative distribution functions (i.e.
pweibull(x, wshape, wscale)
and
pgpd(x, u, sigmau, xi)
) respectively.
The cumulative distribution function for pre-specified , upto the
threshold
, is given by:
and above the threshold :
Notice that these definitions are equivalent when .
The Weibull is defined on the non-negative reals, so the threshold must be positive.
See gpd
for details of GPD upper tail component and
dweibull
for details of weibull bulk component.
Value
dweibullgpd
gives the density,
pweibullgpd
gives the cumulative distribution function,
qweibullgpd
gives the quantile function and
rweibullgpd
gives a random sample.
Note
All inputs are vectorised except log
and lower.tail
.
The main inputs (x
, p
or q
) and parameters must be either
a scalar or a vector. If vectors are provided they must all be of the same length,
and the function will be evaluated for each element of vector. In the case of
rweibullgpd
any input vector must be of length n
.
Default values are provided for all inputs, except for the fundamentals
x
, q
and p
. The default sample size for
rweibullgpd
is 1.
Missing (NA
) and Not-a-Number (NaN
) values in x
,
p
and q
are passed through as is and infinite values are set to
NA
. None of these are not permitted for the parameters.
Error checking of the inputs (e.g. invalid probabilities) is carried out and will either stop or give warning message as appropriate.
Author(s)
Yang Hu and Carl Scarrott carl.scarrott@canterbury.ac.nz
References
http://en.wikipedia.org/wiki/Weibull_distribution
http://en.wikipedia.org/wiki/Generalized_Pareto_distribution
Scarrott, C.J. and MacDonald, A. (2012). A review of extreme value threshold estimation and uncertainty quantification. REVSTAT - Statistical Journal 10(1), 33-59. Available from http://www.ine.pt/revstat/pdf/rs120102.pdf
Behrens, C.N., Lopes, H.F. and Gamerman, D. (2004). Bayesian analysis of extreme events with threshold estimation. Statistical Modelling. 4(3), 227-244.
See Also
Other weibullgpd: fitmweibullgpd
,
fweibullgpdcon
, fweibullgpd
,
itmweibullgpd
, weibullgpdcon
Other weibullgpdcon: fweibullgpdcon
,
fweibullgpd
, itmweibullgpd
,
weibullgpdcon
Other itmweibullgpd: fitmweibullgpd
,
fweibullgpdcon
, fweibullgpd
,
itmweibullgpd
, weibullgpdcon
Other fweibullgpd: fweibullgpd
Examples
## Not run:
set.seed(1)
par(mfrow = c(2, 2))
x = rweibullgpd(1000)
xx = seq(-1, 6, 0.01)
hist(x, breaks = 100, freq = FALSE, xlim = c(-1, 6))
lines(xx, dweibullgpd(xx))
# three tail behaviours
plot(xx, pweibullgpd(xx), type = "l")
lines(xx, pweibullgpd(xx, xi = 0.3), col = "red")
lines(xx, pweibullgpd(xx, xi = -0.3), col = "blue")
legend("topleft", paste("xi =",c(0, 0.3, -0.3)),
col=c("black", "red", "blue"), lty = 1)
x = rweibullgpd(1000, phiu = 0.2)
hist(x, breaks = 100, freq = FALSE, xlim = c(-1, 6))
lines(xx, dweibullgpd(xx, phiu = 0.2))
plot(xx, dweibullgpd(xx, xi=0, phiu = 0.2), type = "l")
lines(xx, dweibullgpd(xx, xi=-0.2, phiu = 0.2), col = "red")
lines(xx, dweibullgpd(xx, xi=0.2, phiu = 0.2), col = "blue")
legend("topleft", c("xi = 0", "xi = 0.2", "xi = -0.2"),
col=c("black", "red", "blue"), lty = 1)
## End(Not run)