Autoregressive models for maxima and their applications to CH4and N2O

Abstract Recordings of daily, weekly, or yearly maxima in environmental time series are classically fitted by the generalized extreme value (GEV) distribution that originates from the well‐established extreme value theory (EVT). One special case of such GEV distribution is the Gumbel family which corresponds to the modeling of maxima stemming from light‐tailed distributions. To capture temporal dependencies, linear autoregressive (AR) processes offer a simple and elegant framework. Our objective is to extend linear AR models in such a way that they handle Gumbel distributed maxima. To reach this goal, we take advantage of the stability of Gumbel random variables when added to the logarithm of a positive α‐stable random variable. This allows us to propose a linear Gumbel distributed AR model whose main theoretical properties are derived. For the atmospheric scientist, this link between linear AR processes and EVT widens the statistical treatment of extreme environmental recordings in which temporal dependencies are present. For example, our model is fitted to daily and weekly maxima of methane (CH4) and daily maxima of nitrous oxide (N2O) measured in Gif‐sur‐Yvette (France). Simulation results are also presented in order to assess the quality of our parameter estimations for finite samples. Copyright © 2009 John Wiley & Sons, Ltd.