B1763
Title: Quantile-based approximation and decomposition of the Cramer distance
Authors: Johannes Resin - Heidelberg University (Germany) [presenting]
Timo Dimitriadis - Heidelberg University (Germany)
Johannes Bracher - Karlsruhe Institute of Technology (Germany)
Daniel Wolffram - Karlsruhe Institute of Technology (Germany)
Abstract: The Cramer distance (CD), also referred to as the integrated squared distance, is a commonly used distance between probability distributions. In the context of probabilistic forecasting, it can be used both to assess the similarity between different forecast distributions and to compare a posited distribution with the empirical distribution of a sample. We investigate a quantile-based representation of the CD, which is useful in two ways. Firstly, the representation gives rise to a quantile-based approximation of the CD, which can be used if forecast distributions are provided as quantiles at pre-specified levels and have the desirable property of being a k-proper divergence. Secondly, the alternative representation can be decomposed into four components, which capture shifts and differences in dispersion between the two distributions. The merits of the quantile-based approximation and its decomposition are demonstrated in applications from climatology, epidemiology and economics.
