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,...,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(x, omega)

Arguments

x Sample of strictly increasing observations.
omega Known endpoint. Make sure that omega >= X_{(n)}.

Value

n x 3 matrix with columns: indices k, Falk's MVUE estimator using the smoothing method, and the ordinary Falk MVUE estimator based on the order statistics.

Author(s)

Kaspar Rufibach (maintainer), kaspar.rufibach@gmail.com

Samuel Mueller, mueller@maths.uwa.edu.au,
http://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. (2006). Smooth tail index estimation. J. Stat. Comput. Simul., to appear.

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)

# compute tail index estimators
omega <- -1 / gam
falkMVUE(x, omega)

[Package smoothtail version 1.1.2 Index]