falkMVUE: 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(x, omega)
Arguments
x
Sample of strictly increasing observations.
omega
Known endpoint. Make sure that $\omega \ge X_{(n)}$.
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.
References
Mueller, S. and Rufibach K. (2006).
Smooth tail index estimation.
Preprint.
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.