A parametric model for distributions with flexible behavior in both tails

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.