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@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
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.
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
.
# generate ordered random sample from GPD set.seed(1977) n <- 20 gam <- -0.75 x <- rgpd(n, gam) # compute tail index estimators pickands(x)