Parametric models for distributions when interest is in extremes with an application to daily temperature

Abstract

When making inferences about extreme quantiles, using simple parametric models for the entire distribution can be problematic in that a model that accurately describes the bulk of the distribution may lead to substantially biased estimates of extreme quantiles if the model is misspecified. One way to address this problem is to use flexible parametric families of distributions. For the setting where extremes in both the upper and lower tails are of interest, this paper describes various approaches to quantifying notions of flexibility and then proposes new parametric classes of distributions that satisfy these notions and are computable without requiring numerical integration. A semiparametric extension of these distributions is proposed when the parametric classes are not sufficiently flexible. Some of the new models are applied to daily temperature in July from an ensemble of 50 climate model runs that can be treated as independent realizations of the climate system over the period studied. The large ensemble makes it possible to compare estimates of extreme quantiles based on a single model run to estimates based on the full ensemble. For these data, at the four largest US cities, Chicago, Houston, Los Angeles and New York City, the parametric models generally dominate estimates based on fitting generalized Pareto distributions to some fraction of the most extreme observations, sometimes by a substantial margin. Thus, in at least this setting, parametric models not only provide a way to estimate the whole distribution, they also result in better estimates of extreme quantiles than traditional extreme value approaches.