# falkMVUE: Compute original and smoothed version of Falk's estimator for... In smoothtail: Smooth Estimation of GPD Shape Parameter

## 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 falkMVUE(est, omega, ks = NA) 

## Arguments

 est Log-concave density estimate based on the sample as output by logConDens (a dlc object). omega Known endpoint. Make sure that ω ≥ X_{(n)}. ks Indices k at which Falk's estimate should be computed. If set to NA defaults to 2, …, n-1.

## 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

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.

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.
  1 2 3 4 5 6 7 8 9 10 11 12 # generate ordered random sample from GPD set.seed(1977) n <- 20 gam <- -0.75 x <- rgpd(n, gam) ## generate dlc object est <- logConDens(x, smoothed = FALSE, print = FALSE, gam = NULL, xs = NULL) # compute tail index estimators omega <- -1 / gam falkMVUE(est, omega)