# PickandsEstimator

0th

Percentile

##### Function to compute Pickands estimates for the GPD and GEVD

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

Keywords
univar
##### 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

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 $0.023$.

##### References

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

##### Aliases
• PickandsEstimator
• .PickandsEstimator
##### Examples
# NOT RUN {
## (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)
# }

Documentation reproduced from package RobExtremes, version 1.2.0, License: LGPL-3

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