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, \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 |
ks |
Indices |
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), kaspar.rufibach@gmail.com,
http://www.kasparrufibach.ch
Samuel Mueller, samuel.mueller@sydney.edu.au,
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)