PickandsEstimator: Function to compute Pickands estimates for the GPD and GEVD

In RobExtremes: Optimally Robust Estimation for Extreme Value Distributions

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 \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 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 \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.

Author(s)

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

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.

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)

Example output

Loading required package: distrMod
Loading required package: distr
Loading required package: startupmsg
:startupmsg>  Utilities for Start-Up Messages (version 0.9.6)
:startupmsg> 
:startupmsg>  For more information see ?"startupmsg",
:startupmsg>  NEWS("startupmsg")

Loading required package: sfsmisc
:distr>  Object Oriented Implementation of Distributions (version
:distr>  2.8.0)
:distr> 
:distr>  Attention: Arithmetics on distribution objects are
:distr>  understood as operations on corresponding random variables
:distr>  (r.v.s); see distrARITH().
:distr> 
:distr>  Some functions from package 'stats' are intentionally masked
:distr>  ---see distrMASK().
:distr> 
:distr>  Note that global options are controlled by distroptions()
:distr>  ---c.f. ?"distroptions".
:distr> 
:distr>  For more information see ?"distr", NEWS("distr"), as well as
:distr>    http://distr.r-forge.r-project.org/
:distr>  Package "distrDoc" provides a vignette to this package as
:distr>  well as to several extension packages; try
:distr>  vignette("distr").


Attaching package: ‘distr’

The following objects are masked from ‘package:stats’:

    df, qqplot, sd

Loading required package: distrEx
:distrEx>  Extensions of Package 'distr' (version 2.8.0)
:distrEx> 
:distrEx>  Note: Packages "e1071", "moments", "fBasics" should be
:distrEx>  attached /before/ package "distrEx". See
:distrEx>  distrExMASK().Note: Extreme value distribution
:distrEx>  functionality has been moved to
:distrEx> 
:distrEx>        package "RobExtremes". See distrExMOVED().
:distrEx> 
:distrEx>  For more information see ?"distrEx", NEWS("distrEx"), as
:distrEx>  well as
:distrEx>    http://distr.r-forge.r-project.org/
:distrEx>  Package "distrDoc" provides a vignette to this package
:distrEx>  as well as to several related packages; try
:distrEx>  vignette("distr").


Attaching package: ‘distrEx’

The following objects are masked from ‘package:stats’:

    IQR, mad, median, var

Loading required package: RandVar
:RandVar>  Implementation of Random Variables (version 1.2.1)
:RandVar> 
:RandVar>  For more information see ?"RandVar", NEWS("RandVar"), as
:RandVar>  well as
:RandVar>    http://robast.r-forge.r-project.org/
:RandVar>  This package also includes a vignette; try
:RandVar>  vignette("RandVar").

Loading required package: MASS
Loading required package: stats4
:distrMod>  Object Oriented Implementation of Probability Models
:distrMod>  (version 2.8.4)
:distrMod> 
:distrMod>  Some functions from pkg's 'base' and 'stats' are
:distrMod>  intentionally masked ---see distrModMASK().
:distrMod> 
:distrMod>  Note that global options are controlled by
:distrMod>  distrModoptions() ---c.f. ?"distrModoptions".
:distrMod> 
:distrMod>  For more information see ?"distrMod",
:distrMod>  NEWS("distrMod"), as well as
:distrMod>    http://distr.r-forge.r-project.org/
:distrMod>  There is a vignette to this package; try
:distrMod>  vignette("distrMod").
:distrMod>  Package "distrDoc" provides a vignette to the other
:distrMod>  distrXXX packages,
:distrMod>  as well as to several related packages; try
:distrMod>  vignette("distr").


Attaching package: ‘distrMod’

The following object is masked from ‘package:stats4’:

    confint

The following object is masked from ‘package:stats’:

    confint

The following object is masked from ‘package:base’:

    norm

Loading required package: ROptEst
Loading required package: RobAStBase
Loading required package: rrcov
Loading required package: robustbase
Scalable Robust Estimators with High Breakdown Point (version 1.5-5)

:RobAStBase>  Robust Asymptotic Statistics (version 1.2.1)
:RobAStBase> 
:RobAStBase>  Some functions from pkg's 'stats' and 'graphics'
:RobAStBase>  are intentionally masked ---see RobAStBaseMASK().
:RobAStBase> 
:RobAStBase>  Note that global options are controlled by
:RobAStBase>  RobAStBaseoptions() ---c.f. ?"RobAStBaseoptions".
:RobAStBase> 
:RobAStBase>  For more information see ?"RobAStBase",
:RobAStBase>  NEWS("RobAStBase"), as well as
:RobAStBase>    http://robast.r-forge.r-project.org/


Attaching package: ‘RobAStBase’

The following object is masked from ‘package:graphics’:

    clip

Loading required package: evd
:RobExtremes>  Optimally Robust Estimation for Extreme Value
:RobExtremes>  Distributions (version 1.2.0)
:RobExtremes> 
:RobExtremes> 
:RobExtremes>  For more information see ?"RobExtremes",
:RobExtremes>  NEWS("RobExtremes"), as well as
:RobExtremes>    http://robast.r-forge.r-project.org/


Attaching package: ‘RobExtremes’

The following objects are masked from ‘package:robustbase’:

    Qn, Sn

Evaluations of PickandsEstimator:
---------------------------------
An object of class “Estimate” 
generated by call
  PickandsEstimator(x = x, ParamFamily = P)
samplesize:   50
estimate:
     scale       shape  
  0.5965601   2.6028314 
 (0.3762822) (0.9050787)
asymptotic (co)variance (multiplied with samplesize):
           scale     shape
scale   7.079416 -12.26167
shape -12.261670  40.95837
Infos:
     method              message
[1,] "PickandsEstimator" ""     
Evaluations of PickandsEstimator:
---------------------------------
An object of class “Estimate” 
generated by call
  PickandsEstimator(x = y, ParamFamily = G)
samplesize:   50
estimate:
     scale       shape  
  0.2180708   2.7928952 
 (0.2214988) (1.0254982)
asymptotic (co)variance (multiplied with samplesize):
          scale     shape
scale  2.453087 -7.936753
shape -7.936753 52.582325
Infos:
     method              message
[1,] "PickandsEstimator" ""     

RobExtremes documentation built on May 2, 2019, 3:44 p.m.