smoothtail (version 2.0.4)

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(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 $3, \ldots, n-1$.

Value

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

References

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