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 γ \in [-1,0]. Precisely, for k=4, …, n
\hat γ^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.
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Log-concave density estimate based on the sample as output by |
ks |
Indices k at which Falk's estimate should be computed. If set to |
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.
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
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.
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 falk
, falkMVUE
, generalizedPick
.
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