Statistical Distances and the Construction of Evidence Functions for Model Adequacy

Abstract

Over the past years, distances and divergences have been extensively used not only in the statistical literature or in probability and information theory, but also in other scientific areas such as engineering, machine learning, biomedical sciences, as well as ecology. Statistical distances, viewed either as building blocks of evidence generation or as evidence generation vehicles in themselves, provide a natural way to create a global framework for inference in parametric and semiparametric models. More precisely, quadratic distance measures play an important role in goodness-of-fit tests, estimation, prediction or model selection. Provided that specific properties are fulfilled, alternative statistical distances (or divergences) can effectively be used to construct evidence functions. In the present article, we discuss an intrinsic approach to the notion of evidence and present a brief literature review related to its interpretation. We examine several statistical distances, both quadratic and non-quadratic, and their properties in relation to important aspects of evidence generation. We provide an extensive description of their role in model identification and model assessment. Further, we introduce an explanatory plot that is based on quadratic distances to visualize the strength of evidence provided by the ratio of standardized quadratic distances and exemplify its use. In this setting, emphasis is placed on determining the sense in which we can provide meaningful interpretations of the distances as measures of statistical loss. We conclude by summarizing the main contributions of this work.