Compute generalized Pickand's estimator

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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 Segers' estimator of the shape parameter γ, see Segers (2005). Precisely, for k = \{1, …, n-1\}, the estimator can be written as

\hat γ^k_{\rm{Segers}}(H) = ∑_{j=1}^k \Bigl(λ(j/k) - λ((j-1)/k)\Bigr) \log \Bigl(H^{-1}((n-\lfloor cj \rfloor)/n)-H^{-1}((n-j)/n) \Bigr)

for H either the empirical or the distribution function based on the log–concave density estimator and λ the mixing measure given in Segers (2005), Theorem 4.1, (i). Note that for any k, \hat γ^k_{\rm{Segers}} : R^n \to (-∞, ∞). If \hat γ_{\rm{Segers}} \not \in [-1,0), then it is likely that the log-concavity assumption is violated.

Usage

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generalizedPick(est, c, gam0, ks = NA)

Arguments

est

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

c

Number in (0,1), determining the spacings that are used.

gam0

Number in R \setminus 0.5, specifying the mixing measure.

ks

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

Value

n x 3 matrix with columns: indices k, Segers' estimator using the smoothing method, and the ordinary Segers' 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.

Segers, J. (2005). Generalized Pickands estimators for the extreme value index. J. Statist. Plann. Inference, 128, 381–396.

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, falkMVUE.

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
generalizedPick(est, c = 0.75, gam0 = -1/3)