generalizedPick {smoothtail}R Documentation

Compute generalized Pickand's estimator

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 Segers' estimator of the shape parameter gamma, see Segers (2005). Precisely, for k = {1, ..., n-1}, the estimator can be written as

hat gamma^k_{rm{Segers}}(H) = sum_{j=1}^k Bigl(λ(j/k) - λ((j-1)/k)Bigr) log Bigl(H^{-1}((n-lfloor cj rfloor)/n)-H^{-1}((n-j)/n) Bigr)

for H either the empirical or the distribution function based on the log–concave density estimator and λ the mixing measure given in Segers (2005), Theorem 4.1, (i). Note that for any k, hat gamma^k_{rm{Segers}} : R^n to (-infty, infty). If hat gamma_{rm{Segers}} not in [-1,0), then it is likely that the log-concavity assumption is violated.

Usage

generalizedPick(x, c, gam0)

Arguments

x Sample of strictly increasing observations.
c Number in (0,1), determining the spacings that are used.
gam0 Number in R setminus 0.5, specifying the mixing measure.

Value

n x 3 matrix with columns: indices k, Segers' estimator using the smoothing method, and the ordinary Segers' 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.

Segers, J. (2005). Generalized Pickands estimators for the extreme value index. J. Statist. Plann. Inference, 128, 381–396.

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 pickands, 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
genPick.logcon <- generalizedPick(x, c = 0.75, gam0 = -1/3)

[Package smoothtail version 1.1.2 Index]