pickands {smoothtail} | R Documentation |
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.
pickands(x)
x |
Sample of strictly increasing observations. |
n x 3 matrix with columns: indices k, Pickands' estimator using the smoothing method, and the ordinary Pickands' estimator based on the order statistics.
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
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.
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
.
# generate ordered random sample from GPD set.seed(1977) n <- 20 gam <- -0.75 x <- rgpd(n, gam) # compute tail index estimators pickands(x)