Abstract
Extreme value theory motivates estimating extreme upper quantiles of a distribution by selecting some threshold, discarding those observations below the threshold and fitting a generalized Pareto distribution to exceedances above the threshold via maximum likelihood. This sharp cutoff between observations that are used in the parameter estimation and those that are not is at odds with statistical practice for analogous problems such as nonparametric density estimation, in which observations are typically smoothly downweighted as they become more distant from the value at which the density is being estimated. By exploiting the fact that the order statistics of independent and identically distributed observations form a Markov chain, this work shows how one can obtain a natural weighted composite log-likelihood function for fitting generalized Pareto distributions to exceedances over a threshold. A method for producing confidence intervals based on inverting a test statistic calibrated via parametric bootstrapping is proposed. Some theory demonstrates the asymptotic advantages of using weights in the special case when the shape parameter of the limiting generalized Pareto distribution is known to be 0. Methods for extending this approach to observations that are not identically distributed are described and applied to an analysis of daily precipitation data in New York City. Perhaps the most important practical finding is that including weights in the composite log-likelihood function can reduce the sensitivity of estimates to small changes in the threshold.
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The precipitation data is available from the National Centers for Environmental Information at https://www.ncdc.noaa.gov/cdo-web/datasets/GHCND/stations/GHCND:USW00094728/detail.
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Acknowledgements
The author would like to thank the Associate Editor and referees for their many helpful comments on the substance and presentation of this work and Mitchell Krock for providing the software to carry out the Minneapolis temperature simulations.
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This material was based upon work supported by the U.S. Department of Energy, Office of Science, Office of Advanced Scientific Computing Research (ASCR) under Contract DE-AC02-06CH11347.
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Stein, M.L. A weighted composite log-likelihood approach to parametric estimation of the extreme quantiles of a distribution. Extremes 26, 469–507 (2023). https://doi.org/10.1007/s10687-023-00466-w
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DOI: https://doi.org/10.1007/s10687-023-00466-w
Keywords
- Extreme value theory
- Generalized Pareto distribution
- Order statistics
- Kernel density estimation
- Test inversion bootstrapping