Abstract
The two parametric distribution functions appearing in the extreme-value theory - the generalized extremevalue distribution and the generalized Pareto distribution - have log-concave densities if the extreme-value index γ ε [-1,0]. Replacing the order statistics in tail-index estimators by their corresponding quantiles from the distribution function that is based on the estimated log-concave density f̂n leads to novel smooth quantile and tail-index estimators. These new estimators aim at estimating the tail index especially in small samples. Acting as a smoother of the empirical distribution function, the log-concave distribution function estimator reduces estimation variability to a much greater extent than it introduces bias. As a consequence, Monte Carlo simulations demonstrate that the smoothed version of the estimators are well superior to their non-smoothed counterparts, in terms of mean-squared error.
Original language | English |
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Pages (from-to) | 1155-1167 |
Number of pages | 13 |
Journal | Journal of Statistical Computation and Simulation |
Volume | 79 |
Issue number | 9 |
DOIs | |
Publication status | Published - Sept 2009 |
Externally published | Yes |
Keywords
- 'extreme-value' theory
- log-concave density estimation
- negative Hill estimator
- Pickands estimator
- tail-index estimation
- small-sample performance