falk {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 Falk's estimator of the shape parameter gamma in [-1,0]. Precisely,
hat gamma_{rm{Falk}} = hat gamma_{rm{Falk}}(k, n) = frac{1}{k-1} sum_{j=2}^k log Bigl(frac{X_{(n)}-H^{-1}((n-j+1)/n)}{X_{(n)}-H^{-1}((n-k)/n)} Bigr), ; ; k=3, ... ,n-1
for $H$ either the empirical or the distribution function based on the log–concave density estimator. Note that for any k, hat gamma_{rm{Falk}} : R^n to (-infty, 0). If hat gamma_{rm{Falk}} not in [-1,0), then it is likely that the log-concavity assumption is violated.
falk(x)
x |
Sample of strictly increasing observations. |
n x 3 matrix with columns: indices k, Falk's estimator using the smoothing method, and the ordinary Falk's 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.
Falk, M. (1995). Some best parameter estimates for distributions with finite endpoint. Statistics, 27, 115–125.
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
, falkMVUE
.
# generate ordered random sample from GPD set.seed(1977) n <- 20 gam <- -0.75 x <- rgpd(n, gam) # compute tail index estimators falk(x)