# Compute generalized Pickand's estimator

### Description

Given an ordered sample of either exceedances or upper order statistics which is to be modeled using a GPD with
distribution function *F*, this function provides Segers' estimator of the shape parameter *γ*,
see Segers (2005). Precisely, for *k = \{1, …, n-1\}*, the estimator can be written as

*\hat γ^k_{\rm{Segers}}(H) = ∑_{j=1}^k \Bigl(λ(j/k) - λ((j-1)/k)\Bigr) \log \Bigl(H^{-1}((n-\lfloor cj \rfloor)/n)-H^{-1}((n-j)/n) \Bigr)*

for *H* either the empirical or the distribution function based on the log–concave density estimator
and *λ* the mixing measure given in Segers (2005), Theorem 4.1, (i).
Note that for any *k*, *\hat γ^k_{\rm{Segers}} : R^n \to (-∞, ∞)*.
If *\hat γ_{\rm{Segers}} \not \in [-1,0)*, then it is likely that the log-concavity assumption is violated.

### Usage

1 | ```
generalizedPick(est, c, gam0, ks = NA)
``` |

### Arguments

`est` |
Log-concave density estimate based on the sample as output by |

`c` |
Number in |

`gam0` |
Number in |

`ks` |
Indices |

### Value

n x 3 matrix with columns: indices *k*, Segers' estimator using the smoothing method, and
the ordinary Segers' estimator based on the order statistics.

### Author(s)

Kaspar Rufibach (maintainer), kaspar.rufibach@gmail.com,

http://www.kasparrufibach.ch

Samuel Mueller, samuel.mueller@sydney.edu.au,

www.maths.usyd.edu.au/ut/people?who=S_Mueller

Kaspar Rufibach acknowledges support by the Swiss National Science Foundation SNF, http://www.snf.ch

### References

Mueller, S. and Rufibach K. (2009).
Smooth tail index estimation.
*J. Stat. Comput. Simul.*, **79**, 1155–1167.

Segers, J. (2005).
Generalized Pickands estimators for the extreme value index.
*J. Statist. Plann. Inference*, **128**, 381–396.

### See Also

Other approaches to estimate *γ* based on the fact that the density is log–concave, thus
*γ \in [-1,0]*, are available as the functions `pickands`

, `falk`

, `falkMVUE`

.

### Examples

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