pickands {smoothtail}

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, \ldots, 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(est, ks = NA)

Arguments

est

Log-concave density estimate based on the sample as output by logConDens (a dlc object).

ks

Indices k at which Falk's estimate should be computed. If set to NA defaults to 4, \ldots, n.

Value

n x 3 matrix with columns: indices k, Pickands' estimator using the log-concave density estimate, and the ordinary Pickands' estimator based on the order statistics.

Author(s)

Kaspar Rufibach (maintainer), [e-mail],
http://www.kasparrufibach.ch

Samuel Mueller, [e-mail],
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.

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)

## generate dlc object
est <- logConDens(x, smoothed = FALSE, print = FALSE, gam = NULL, xs = NULL)

# compute tail index estimators
pickands(est)