Return level bounds for discrete and continuous random variables

For a wide range of applications in hydrology and climate studies, the return level is a fundamental quantity to build dykes, propose flood planning policies, and study weather and climate phenomena linked to the behavior of the upper tail of a distribution. More precisely, z (t) is called the return level associated with a given return period t if the level z (t) is expected to be exceeded on average once every t years. To estimate this level in the independent and identically distributed setting, Extreme Value Theory (EVT) has been classically used by assuming that exceedances above a high threshold approximately follow a Generalized Pareto distribution (GPD). This approximation is based on an asymptotic argument, but the rate of convergence may be slow in some cases, e.g., Gaussian variables, and the choice of an appropriate threshold difficult. As an alternative, we propose and study simple estimators of lower and upper return level bounds. This approach has several advantages. It works for both small and moderate sample sizes and for discrete and continuous random variables. In addition, no threshold selection procedure is needed. Still, there is a clear link with EVT because our bounds can be viewed as extensions of the concept of the probability weighted moments that has been classically used in hydrology. In particular, some moment conditions have to be satisfied in order to derive the asymptotic properties of our estimators. We apply our methodology to a few simulations and two climate data sets.