Function `PickandsEstimator`

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

.

1 2 3 4 5 | ```
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
*\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 Nataliya.Horbenko@itwm.fraunhofer.de,

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`

1 2 3 4 5 6 7 8 9 10 | ```
## (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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