generalizedPick {smoothtail} | R Documentation |
Compute generalized Pickand's estimator
Description
Given an ordered sample of either exceedances or upper order statistics which is to be modeled using a GPD with
distribution function F
, this function provides Segers' estimator of the shape parameter \gamma
,
see Segers (2005). Precisely, for k = \{1, \ldots, n-1\}
, the estimator can be written as
\hat \gamma^k_{\rm{Segers}}(H) = \sum_{j=1}^k \Bigl(\lambda(j/k) - \lambda((j-1)/k)\Bigr) \log \Bigl(H^{-1}((n-\lfloor cj \rfloor)/n)-H^{-1}((n-j)/n) \Bigr)
for H
either the empirical or the distribution function based on the log–concave density estimator
and \lambda
the mixing measure given in Segers (2005), Theorem 4.1, (i).
Note that for any k
, \hat \gamma^k_{\rm{Segers}} : R^n \to (-\infty, \infty)
.
If \hat \gamma_{\rm{Segers}} \not \in [-1,0)
, then it is likely that the log-concavity assumption is violated.
Usage
generalizedPick(est, c, gam0, ks = NA)
Arguments
est |
Log-concave density estimate based on the sample as output by |
c |
Number in |
gam0 |
Number in |
ks |
Indices |
Value
n x 3 matrix with columns: indices k
, Segers' estimator using the smoothing method, and
the ordinary Segers' 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.
Segers, J. (2005). Generalized Pickands estimators for the extreme value index. J. Statist. Plann. Inference, 128, 381–396.
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
, falk
, falkMVUE
.
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
generalizedPick(est, c = 0.75, gam0 = -1/3)