falk {smoothtail} | R Documentation |
Compute original and smoothed version of Falk's estimator
Description
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, \ldots ,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.
Usage
falk(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
, Falk's estimator based on the log-concave density estimate, and
the ordinary Falk's 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.
Falk, M. (1995). Some best parameter estimates for distributions with finite endpoint. Statistics, 27, 115–125.
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 pickands
, 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 estimator
falk(est)