pickands {smoothtail}R Documentation

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 gamma in [-1,0]. Precisely, for k=4, ..., n

hat gamma^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

pickands(x)

Arguments

x Sample of strictly increasing observations.

Value

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

Author(s)

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

Samuel Mueller, mueller@maths.uwa.edu.au,
http://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. (2006). Smooth tail index estimation. J. Stat. Comput. Simul., to appear.

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

See Also

Other approaches to estimate gamma based on the fact that the density is log–concave, thus gamma in [-1,0], are available as the functions falk, falkMVUE, generalizedPick.

Examples

# generate ordered random sample from GPD
set.seed(1977)
n <- 20
gam <- -0.75
x <- rgpd(n, gam)

# compute tail index estimators
pickands(x)

[Package smoothtail version 1.1.2 Index]