A parametric model for distributions with flexible behavior in both tails

Abstract

Summary For many problems of inference about a marginal distribution function, while the entire distribution is important, extreme quantiles are of particular interest because rare outcomes may have large consequences. In some applications, only the extreme upper quantiles require extra attention, but in, for example, climatological applications, extremes in both tails of the distribution can be impactful. A possible approach in this setting is to use parametric families of distributions that have flexible behavior in both tails. One way to quantify this property is to require that, for any two generalized Pareto distributions, there is a member of the parametric family that behaves like one of the generalized Pareto distributions in the upper tail and like the negative of the other generalized Pareto distribution in the lower tail. This work proposes some specific quantifications of this notion and describes parametric families of distributions that satisfy these specifications. The proposed families all have closed form expressions for their densities and, hence, are convenient for use in practice. A simulation study shows how one of the proposed families can work well for estimating all quantiles when both tails of a distribution are heavy tailed. An application to climate model output shows this family can also work well when applied to daily average January temperature near Calgary, for which the evolving distribution over time due to climate change is difficult to model accurately by any standard parametric family.

Nonstationary seasonal model for daily mean temperature distribution bridging bulk and tails

“…A recent model proposed by Stein (2020) fits the entire distribution and has flexible behavior in both tails. Other works have also attempted to bridge the tails and bulk of a distribution; see Scarrott and MacDonald (2012) for a review.…”

“…Huang et al (2019b) suggest a semiparametric approach incorporating log-histosplines. A number of the limitations of these approaches, such as flexible behavior in only one tail, restrictions to positive or heavy-tailed variables, or the need to numerically compute a normalizing constant, are obviated by the approach of Stein (2020), which provides a comprehensive approach to handle the bulk and both tails of a distribution.…”

An Extreme Value Bayesian Lasso for the Conditional Left and Right Tails

“…Moreover, in many situations of applied interest, it would be desirable to model both the lower values of the data along with the extreme values in a regression framework. Our model builds over Papastathopoulos and Tawn (2013) and Naveau et al (2016) who proposed an extended generalized Pareto distribution (EGPD) to jointly model low, moderate, and extreme observations-without the need of threshold selection; other interesting options for modeling both the bulk and the tail of a distribution, include extreme value mixture models (e.g., Frigessi et al 2002, Behrens et al 2004, Carreau and Bengio 2009, Cabras and Castellanos 2011, MacDonald et al 2011, do Nascimento et al 2012 as well as composition-based approaches (Stein 2020(Stein , 2021.…”

Spatial hierarchical modeling of threshold exceedances using rate mixtures

“…With subasymptotic tail models, parameter estimates are usually less sensitive to the threshold, the choice of which then becomes less crucial for inference, and we can thus set lower thresholds. In some approaches (see, e.g., Naveau et al, 2016; Stein, 2020a, 2020b), the threshold choice is even bypassed by adding parameters that provide separate control over bulk properties of the distribution, such that models are expected to provide a satisfactory fit of the tail, even if the whole sample is used for estimation.…”