smoothtail (version 1.0)

falk: 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(x)

Arguments

x

Sample of strictly increasing observations.

Value

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

References

Mueller, S. and Rufibach K. (2006). Smooth tail index estimation. Preprint. 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.

Examples

# generate ordered random sample from GPD
set.seed(1977)
n <- 20
gam <- -0.75
x <- rgpd(n, gam)

# compute tail index estimators
falk(x)