smoothtail (version 2.0.4)

pickands: Compute original and smoothed version of Pickands' estimator

Description

Given an ordered sample of either exceedances or upper order statistics which is to be modeled using a GPD, this function provides Pickands' estimator of the shape parameter $\gamma \in [-1,0]$. Precisely, for $k=4, \ldots, n$ $$\hat \gamma^k_{\rm{Pick}} = \frac{1}{\log 2} \log \Bigl(\frac{H^{-1}((n-r_k(H)+1)/n)-H^{-1}((n-2r_k(H) +1)/n)}{H^{-1}((n-2r_k(H) +1)/n)-H^{-1}((n-4r_k(H)+1)/n)} \Bigr)$$ 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(est, ks = NA)

Arguments

est

Log-concave density estimate based on the sample as output by logConDens (a dlc object).

ks

Indices $k$ at which Falk's estimate should be computed. If set to NA defaults to $4, \ldots, n$.

Value

• n x 3 matrix with columns: indices $k$, Pickands' estimator using the log-concave density estimate, and the ordinary Pickands' estimator based on the order statistics.

References

Mueller, S. and Rufibach K. (2009). Smooth tail index estimation. J. Stat. Comput. Simul., 79, 1155--1167. 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, 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
pickands(est)