smoothtail (version 1.0)

pickands: Compute original and smoothed version of Pickand'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 Pickand's estimator of the shape parameter $\gamma \in [-1,0]$. Precisely, for $k=4, \ldots, n$ $$\hat \gamma^k_{\rm{Pick}} = \frac{1}{\log 2} \log \Bigl(\frac{H^{-1}((n-r_k(H)+1)/n)-H^{-1}((n-2r_k(H) +1)/n)}{H^{-1}((n-2r_k(H) +1)/n)-H^{-1}((n-k+1)/n)} \Bigr)$$ for $H$ either the empirical or the distribution function $\hat F_n$ based on the log--concave density estimator and $$r_k(H) = \lfloor k/4 \rfloor$$ if $H$ is the empirical distribution function and $$r_k(H) = k / 4$$ if $H = \hat F_n$.

Usage

pickands(x)

Arguments

x

Sample of strictly increasing observations.

Value

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

References

Mueller, S. and Rufibach K. (2006). Smooth tail index estimation. Preprint. Pickands, J. (1975). Statistical inference using extreme order statistics. Annals of Statistics 3, 119--131.

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 falk, falkMVUE.

Examples

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

# compute tail index estimators
pickands(x)