Abstract: 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 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 talk describes some specific quantifications of this notion and proposes parametric families of distributions that satisfy these specifications.These families all have closed form expressions for their densities and, hence, are convenient for likelihood-based inferences. An application to climate model output shows this family works 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. Time permitting, work by Mitchell Krock on extensions of this model to multivariate distributions will be described.