Affine invariant divergences associated with proper composite scoring rules and their applications

Abstract

In statistical analysis, measuring a score of predictive performance is an important task. In many scientific fields, appropriate scoring rules were tailored to tackle the problems at hand. A proper scoring rule is a popular tool to obtain statistically consistent forecasts. Furthermore, a mathematical characterization of the proper scoring rule was studied. As a result, it was revealed that the proper scoring rule corresponds to a Bregman divergence, which is an extension of the squared distance over the set of probability distributions. In the present paper, we introduce composite scoring rules as an extension of the typical scoring rules in order to obtain a wider class of probabilistic forecasting. Then, we propose a class of composite scoring rules, named Hölder scores, that induce equivariant estimators. The equivariant estimators have a favorable property, implying that the estimator is transformed in a consistent way, when the data is transformed. In particular, we deal with the affine transformation of the data. By using the equivariant estimators under the affine transformation, one can obtain estimators that do no essentially depend on the choice of the system of units in the measurement. Conversely, we prove that the Hölder score is characterized by the invariance property under the affine transformations. Furthermore, we investigate statistical properties of the estimators using Hölder scores for the statistical problems including estimation of regression functions and robust parameter estimation, and illustrate the usefulness of the newly introduced scoring rules for statistical forecasting.