# Compute original and smoothed version of Falk's estimator for a known endpoint

### 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 Falk's estimator of the shape parameter *γ \in [-1,0]*
if the endpoint

*ω(F) = \sup\{x \, : \, F(x) < 1\}*

of *F* is known. Precisely,

*\hat γ_{\rm{MVUE}} = \hat γ_{\rm{MVUE}}(k,n) = \frac{1}{k} ∑_{j=1}^k \log \Bigl(\frac{ω(F)-H^{-1}((n-j+1)/n)}{ω(F)-H^{-1}((n-k)/n)}\Bigr), \; \; k=2,…,n-1*

for *H* either the empirical or the distribution function based on the log–concave density estimator.
Note that for any *k*, *\hat γ_{\rm{MVUE}} : R^n \to (-∞, 0)*. If *\hat γ_{\rm{MVUE}}
\not \in [-1,0)*, then it is likely that the log-concavity assumption is violated.

### Usage

1 |

### Arguments

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

`omega` |
Known endpoint. Make sure that |

`ks` |
Indices |

### Value

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

Falk, M. (1994).
Extreme quantile estimation in *δ*-neighborhoods of generalized Pareto distributions.
*Statistics and Probability Letters*, **20**, 9–21.

Falk, M. (1995).
Some best parameter estimates for distributions with finite endpoint.
*Statistics*, **27**, 115–125.

### 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`

, `generalizedPick`

.

### Examples

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