# PickandsEstimator

##### 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
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

A numeric vector of length `2`

with components
named `scale`

and `shape`

.

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.

##### See Also

##### 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*