PickandsEstimator {RobExtremes} | R Documentation |
Function PickandsEstimator
computes Pickands estimator
(for the GPD and GEVD) at real data and returns an object of class Estimate
.
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)
x |
(empirical) data |
alpha |
numeric > 1; determines the variant of the Pickands-Estimator
based on matching the empirical quantiles to levels
\code{a1=1-1/alpha} and
\code{a2=1-1/alpha^2} (in the GPD case) resp.
\code{a1=exp(-1/alpha)} and
\code{a2=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 |
ParamFamily |
an object of class |
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 |
na.rm |
logical: if |
... |
not yet used. |
GPD.l |
logical: if |
The actual work is done in .PickandsEstimator
.
The wrapper PickandsEstimator
pre-treats the data,
and constructs a respective Estimate
object.
.PickandsEstimator |
A numeric vector of length |
PickandsEstimator |
An object of S4-class |
The scale estimate we use, i.e., with scale = beta and shape = xi, we estimate scale by \code{beta= xi*a1/(alpha^xi-1)}, differs from the one given in the original reference, where it was \code{beta= xi * a1^2 /(a2-2*a1)}. The one chosen here avoids taking differences a2-2*a1 hence does not require a2>2*a1; this leads to (functional) breakdown point (bdp)
min(a1,1-a2,a2-a1)
which is independent xi, whereas the original setting leads to a bdp which is depending on xi
\code{min(a1,1-a2,a2-1+(2*alpha^xi-1)^(-1/xi))} for GPD
\code{min(a1,1-a2,a2-exp(-(2*alpha^xi-1)^(-1/xi)))} 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 a1 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.
Nataliya Horbenko nhorbenko@gmail.com,
Peter Ruckdeschel peter.ruckdeschel@uni-oldenburg.de
P. Ruckdeschel, N. Horbenko (2012): Yet another breakdown point notion:
EFSBP –illustrated at scale-shape models. Metrika, 75(8),
1025–1047.
J. Pickands (1975): Statistical inference using extreme order statistics. Ann. Stat. 3(1), 119–131.
ParamFamily-class
, ParamFamily
,
Estimate-class
## (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)