pickandsplot {evmix} | R Documentation |
Pickands Plot
Description
Produces the Pickand's plot.
Usage
pickandsplot(data, orderlim = NULL, tlim = NULL, y.alpha = FALSE,
alpha = 0.05, ylim = NULL, legend.loc = "topright",
try.thresh = quantile(data, 0.9, na.rm = TRUE),
main = "Pickand's Plot", xlab = "order", ylab = ifelse(y.alpha,
" tail index - alpha", "shape - xi"), ...)
Arguments
data |
vector of sample data |
orderlim |
vector of (lower, upper) limits of order statistics
to plot estimator, or |
tlim |
vector of (lower, upper) limits of range of threshold
to plot estimator, or |
y.alpha |
logical, should shape xi ( |
alpha |
significance level over range (0, 1), or |
ylim |
y-axis limits or |
legend.loc |
location of legend (see |
try.thresh |
vector of thresholds to consider |
main |
title of plot |
xlab |
x-axis label |
ylab |
y-axis label |
... |
further arguments to be passed to the plotting functions |
Details
Produces the Pickand's plot including confidence intervals.
For an ordered iid sequence X_{(1)}\ge X_{(2)}\ge\cdots\ge X_{(n)}
the Pickand's estimator of the reciprocal of the shape parameter \xi
at the k
th order statistic is given by
\hat{\xi}_{k,n}=\frac{1}{\log(2)} \log\left(\frac{X_{(k)}-X_{(2k)}}{X_{(2k)}-X_{(4k)}}\right).
Unlike the Hill estimator it does not assume positive data, is valid for any \xi
and
is location and scale invariant.
The Pickands estimator is defined on orders k=1, \ldots, \lfloor n/4\rfloor
.
Once a sufficiently low order statistic is reached the Pickand's estimator will be constant, upto sample uncertainty, for regularly varying tails. Pickand's plot is a plot of
\hat{\xi}_{k,n}
against the k
. Symmetric asymptotic
normal confidence intervals assuming Pareto tails are provided.
The Pickand's estimator is for the GPD shape \xi
, or the reciprocal of the
tail index \alpha=1/\xi
. The shape is plotted by default using
y.alpha=FALSE
and the tail index is plotted when y.alpha=TRUE
.
A pre-chosen threshold (or more than one) can be given in
try.thresh
. The estimated parameter (\xi
or \alpha
) at
each threshold are plot by a horizontal solid line for all higher thresholds.
The threshold should be set as low as possible, so a dashed line is shown
below the pre-chosen threshold. If Pickand's estimator is similar to the
dashed line then a lower threshold may be chosen.
If no order statistic (or threshold) limits are provided
orderlim = tlim = NULL
then the lowest order statistic is set to X_{(1)}
and
highest possible value X_{\lfloor n/4\rfloor}
. However, Pickand's estimator is always
output for all k=1, \ldots, \lfloor n/4\rfloor
.
The missing (NA
and NaN
) and non-finite values are ignored.
The lower x-axis is the order k
. The upper axis is for the corresponding threshold.
Value
pickandsplot
gives Pickand's plot. It also
returns a dataframe containing columns of the order statistics, order, Pickand's
estimator, it's standard devation and 100(1 - \alpha)\%
confidence
interval (when requested).
Acknowledgments
Thanks to Younes Mouatasim, Risk Dynamics, Brussels for reporting various bugs in these functions.
Note
Asymptotic Wald type CI's are estimated for non-NULL
signficance level alpha
for the shape parameter, assuming exactly GPD tails. When plotting on the tail index scale,
then a simple reciprocal transform of the CI is applied which may well be sub-optimal.
Error checking of the inputs (e.g. invalid probabilities) is carried out and will either stop or give warning message as appropriate.
Author(s)
Carl Scarrott carl.scarrott@canterbury.ac.nz
References
Pickands III, J.. (1975). Statistical inference using extreme order statistics. Annal of Statistics 3(1), 119-131.
Dekkers A. and de Haan, S. (1989). On the estimation of the extreme-value index and large quantile estimation. Annals of Statistics 17(4), 1795-1832.
Resnick, S. (2007). Heavy-Tail Phenomena - Probabilistic and Statistical Modeling. Springer.
See Also
Examples
## Not run:
par(mfrow = c(2, 1))
# Reproduce graphs from Figure 4.7 of Resnick (2007)
data(danish, package="evir")
# Pickand's plot
pickandsplot(danish, orderlim=c(1, 150), ylim=c(-0.1, 2.2),
try.thresh=c(), alpha=NULL, legend.loc=NULL)
# Using default settings
pickandsplot(danish)
## End(Not run)