View source: R/PickandsEstimator.R
PickandsEstimator | 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 |
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
\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
.
Nataliya Horbenko nhorbenko@gmail.com,
Peter Ruckdeschel peter.ruckdeschel@uni-oldenburg.de
Ruckdeschel, P. and Horbenko, N. (2012): Yet another breakdown point notion:
EFSBP –illustrated at scale-shape models. Metrika, 75(8),
1025–1047. \Sexpr[results=rd]{tools:::Rd_expr_doi("10.1007/s00184-011-0366-4")}.
Pickands, J. (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)
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