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.
1 |
est |
Log-concave density estimate based on the sample as output by |
omega |
Known endpoint. Make sure that ω ≥ X_{(n)}. |
ks |
Indices k at which Falk's estimate should be computed. If set to |
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.
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.
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.
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
.
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