PickandsEstimator {RobExtremes}

Function to compute Pickands estimates for the GPD and GEVD

Description

Function PickandsEstimator computes Pickands estimator (for the GPD and GEVD) at real data and returns an object of class Estimate.

Usage

PickandsEstimator(x, ParamFamily=GParetoFamily(), alpha=2,
            name, Infos, nuis.idx = NULL,
            trafo = NULL, fixed = NULL, na.rm = TRUE,
            ...)
.PickandsEstimator(x, alpha=2, GPD.l = TRUE)

Arguments

x

(empirical) data

alpha

numeric > 1; determines the variant of the Pickands-Estimator based on matching the empirical quantiles to levels a_1=1-1/\alpha and a_2=1-1/\alpha^2 (in the GPD case) resp. a_1=\exp(-1/\alpha) and a_1=\exp(-1/\alpha^2) (in the GEVD case) against the population counter parts. The ”classical” Pickands Estimator building up on the median is obtained for alpha=2 for the GPD and for alpha = 1/log(2) for the GEVD. If alpha is missing we set it to the optimal value (see note below).

ParamFamily

an object of class "GParetoFamily" or "GEVFamily".

name

optional name for estimator.

Infos

character: optional informations about estimator

nuis.idx

optionally the indices of the estimate belonging to nuisance parameter

fixed

optionally (numeric) the fixed part of the parameter

trafo

an object of class MatrixorFunction – a transformation for the main parameter

na.rm

logical: if TRUE, the estimator is evaluated at complete.cases(x).

...

not yet used.

GPD.l

logical: if TRUE the variant for GPD is used, else for GEVD.

Details

The actual work is done in .PickandsEstimator. The wrapper PickandsEstimator pre-treats the data, and constructs a respective Estimate object.

Value

.PickandsEstimator

A numeric vector of length 2 with components named scale and shape.

PickandsEstimator

An object of S4-class "Estimate".

Note

The scale estimate we use, i.e., with scale = \beta and shape = \xi, we estimate scale by \beta= \xi a_1/(\alpha^\xi-1), differs from the one given in the original reference, where it was \beta= \xi a_1^2/(a_2-2a_1). The one chosen here avoids taking differences a_2-2a_1 hence does not require a_2 > 2a_1; this leads to (functional) breakdown point (bdp)

\min(a_1,1-a_2,a_2-a_1)

which is independent \xi, whereas the original setting leads to a bdp which is depending on \xi

\min(a_1,1-a_2,a_2-1+(2\alpha^\xi-1)^{-1/\xi})\qquad \mbox{for GPD}

\min(a_1,1-a_2,a_2-\exp(-(2\alpha^\xi-1)^{-1/\xi})) \qquad \mbox{for GEVD}

. As a consequence our setting, the bdp-optimal choice of \alpha for GDP is 2 leading to bdp 1/4, and 2.248 for GEVD leading to bdp 0.180. For comparison, with the original setting, at \xi=0.7, this gives optimal bdp's 0.070 and 0.060 for GPD and GEVD, respectively. The standard choice of \alpha such that a_1 gives the median (\alpha=2 in the GPD and \alpha=1/\log(2) in the GEVD) in our setting gives bdp's of 1/4 and 0.119 for GPD and GEVD, respectively, and in the original setting, at \xi=0.7, gives bdp's 0.064 and 0.023.

Author(s)

Nataliya Horbenko [e-mail],
Peter Ruckdeschel [e-mail]

References

Ruckdeschel, P. and Horbenko, N. (2012): Yet another breakdown point notion: EFSBP –illustrated at scale-shape models. Metrika, 75(8), 1025–1047. doi:10.1007/s00184-011-0366-4.

Pickands, J. (1975): Statistical inference using extreme order statistics. Ann. Stat. 3(1), 119–131.

See Also

ParamFamily-class, ParamFamily, Estimate-class

Examples

## (empirical) Data
set.seed(123)
x <- rgpd(50, scale = 0.5, shape = 3)
y <- rgev(50, scale = 0.5, shape = 3)
## parametric family of probability measures
P <- GParetoFamily(scale = 1, shape = 2)
G <- GEVFamily(scale = 1, shape = 2)
##
PickandsEstimator(x = x, ParamFamily = P)
PickandsEstimator(x = y, ParamFamily = G)