This was part of Climate and Weather Extremes

A weighted composite log-likelihood approach to parametric estimation of the extreme quantiles of a distribution

Michael Stein, Rutgers University

Wednesday, October 5, 2022

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 for fitting generalized Pareto distributions to exceedances over a threshold.  Accurate confidence intervals for extreme quantiles can be obtained by inverting tests based on parametric bootstrapping.  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 can reduce the sensitivity of estimates to small changes in the threshold.