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
the Pickand's estimator of the reciprocal of the shape parameter
at the
th order statistic is given by
Unlike the Hill estimator it does not assume positive data, is valid for any and
is location and scale invariant.
The Pickands estimator is defined on orders
.
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
against the . Symmetric asymptotic
normal confidence intervals assuming Pareto tails are provided.
The Pickand's estimator is for the GPD shape , or the reciprocal of the
tail index
. 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 ( or
) 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 and
highest possible value
. However, Pickand's estimator is always
output for all
.
The missing (NA
and NaN
) and non-finite values are ignored.
The lower x-axis is the order . 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 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)