A Unifying Framework for Some Directed Distances in Statistics

• URL: http://arxiv.org/abs/2203.00863v1
• Date: Wed, 2 Mar 2022 04:24:13 GMT
• Title: A Unifying Framework for Some Directed Distances in Statistics
• Authors: Michel Broniatowski and Wolfgang Stummer
• Abstract summary: Density-based directed distances -- particularly known as divergences -- are widely used in statistics. We provide a general framework which covers in particular both the density-based and distribution-function-based divergence approaches. We deduce new concepts of dependence between random variables, as alternatives to the celebrated mutual information.
• Score: 0.0
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: Density-based directed distances -- particularly known as divergences -- between probability distributions are widely used in statistics as well as in the adjacent research fields of information theory, artificial intelligence and machine learning. Prominent examples are the Kullback-Leibler information distance (relative entropy) which e.g. is closely connected to the omnipresent maximum likelihood estimation method, and Pearson's chisquare-distance which e.g. is used for the celebrated chisquare goodness-of-fit test. Another line of statistical inference is built upon distribution-function-based divergences such as e.g. the prominent (weighted versions of) Cramer-von Mises test statistics respectively Anderson-Darling test statistics which are frequently applied for goodness-of-fit investigations; some more recent methods deal with (other kinds of) cumulative paired divergences and closely related concepts. In this paper, we provide a general framework which covers in particular both the above-mentioned density-based and distribution-function-based divergence approaches; the dissimilarity of quantiles respectively of other statistical functionals will be included as well. From this framework, we structurally extract numerous classical and also state-of-the-art (including new) procedures. Furthermore, we deduce new concepts of dependence between random variables, as alternatives to the celebrated mutual information. Some variational representations are discussed, too.

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