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 γ \in [-1,0] if the endpoint

ω(F) = \sup\{x \, : \, F(x) < 1\}

of F is known. Precisely,

\hat γ_{\rm{MVUE}} = \hat γ_{\rm{MVUE}}(k,n) = \frac{1}{k} ∑_{j=1}^k \log \Bigl(\frac{ω(F)-H^{-1}((n-j+1)/n)}{ω(F)-H^{-1}((n-k)/n)}\Bigr), \; \; k=2,…,n-1

for H either the empirical or the distribution function based on the log–concave density estimator. Note that for any k, \hat γ_{\rm{MVUE}} : R^n \to (-∞, 0). If \hat γ_{\rm{MVUE}} \not \in [-1,0), then it is likely that the log-concavity assumption is violated.

Usage

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falkMVUE(est, omega, ks = NA)

Arguments

est

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

omega

Known endpoint. Make sure that ω ≥ X_{(n)}.

ks

Indices k at which Falk's estimate should be computed. If set to NA defaults to 2, …, n-1.

Value

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

Author(s)

Kaspar Rufibach (maintainer), kaspar.rufibach@gmail.com,
http://www.kasparrufibach.ch

Samuel Mueller, samuel.mueller@sydney.edu.au,
www.maths.usyd.edu.au/ut/people?who=S_Mueller

Kaspar Rufibach acknowledges support by the Swiss National Science Foundation SNF, http://www.snf.ch

References

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

Falk, M. (1994). Extreme quantile estimation in δ-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 γ based on the fact that the density is log–concave, thus γ \in [-1,0], are available as the functions pickands, falk, generalizedPick.

Examples

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# 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
omega <- -1 / gam
falkMVUE(est, omega)