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@freesurf.ch,
http://www.stanford.edu/~kasparr

Samuel Mueller, mueller@maths.uwa.edu.au,
http://www.maths.uwa.edu.au/Members/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. Preprint, available at http://arxiv.org/abs/math.ST/0612140.

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.

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 1.1.1 Index]