falkMVUE {smoothtail} | R Documentation |
Compute original and smoothed version of Falk's estimator for a known endpoint
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 Falk's estimator of the shape parameter \gamma \in [-1,0]
if the endpoint
\omega(F) = \sup\{x \, : \, F(x) < 1\}
of F
is known. Precisely,
\hat \gamma_{\rm{MVUE}} = \hat \gamma_{\rm{MVUE}}(k,n) = \frac{1}{k} \sum_{j=1}^k \log \Bigl(\frac{\omega(F)-H^{-1}((n-j+1)/n)}{\omega(F)-H^{-1}((n-k)/n)}\Bigr), \; \; k=2,\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{MVUE}} : R^n \to (-\infty, 0)
. If \hat \gamma_{\rm{MVUE}}
\not \in [-1,0)
, then it is likely that the log-concavity assumption is violated.
Usage
falkMVUE(est, omega, ks = NA)
Arguments
est |
Log-concave density estimate based on the sample as output by |
omega |
Known endpoint. Make sure that |
ks |
Indices |
Value
n x 3 matrix with columns: indices k
, Falk's MVUE estimator using the log-concave density estimate, and
the ordinary Falk MVUE 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. (1994).
Extreme quantile estimation in \delta
-neighborhoods of generalized Pareto distributions.
Statistics and Probability Letters, 20, 9–21.
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
, falk
, 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 estimators
omega <- -1 / gam
falkMVUE(est, omega)