# Compute original and smoothed version of Pickands' estimator

### Description

Given an ordered sample of either exceedances or upper order statistics which is to be modeled using a GPD, this
function provides Pickands' estimator of the shape parameter *γ \in [-1,0]*.
Precisely, for *k=4, …, n*

*\hat γ^k_{\rm{Pick}} = \frac{1}{\log 2} \log \Bigl(\frac{H^{-1}((n-r_k(H)+1)/n)-H^{-1}((n-2r_k(H) +1)/n)}{H^{-1}((n-2r_k(H) +1)/n)-H^{-1}((n-4r_k(H)+1)/n)} \Bigr)*

for $H$ either the empirical or the distribution function *\hat F_n* based on the log–concave density
estimator and

*r_k(H) = \lfloor k/4 \rfloor*

if *H* is the empirical distribution function and

*r_k(H) = k / 4*

if *H = \hat F_n*.

### Usage

1 |

### Arguments

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

`ks` |
Indices |

### Value

n x 3 matrix with columns: indices *k*, Pickands' estimator using the log-concave density estimate, and
the ordinary Pickands' 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.

Pickands, J. (1975).
Statistical inference using extreme order statistics.
*Annals of Statistics* **3**, 119–131.

### 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 `falk`

, `falkMVUE`

, `generalizedPick`

.

### Examples

1 2 3 4 5 6 7 8 9 10 11 |