# A Unifying Framework for Some Directed Distances in Statistics

arXiv: https://arxiv.org/abs/2203.00863 · HTML: https://mlsanity.com/paper/2203.00863/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 9 of its 193 references that are in the map.
- First posted on arXiv 2 Mar 2022.

## Builds on (most cited first, 9 of 9)

- [Optimizing Variational Representations of Divergences and Accelerating their Statistical Estimation](https://mlsanity.com/paper/2006.08781/) (2020)
- [Information Geometry Connecting Wasserstein Distance and Kullback-Leibler Divergence via the Entropy-Relaxed Transportation Problem](https://mlsanity.com/paper/1709.10219/) (2017)
- [Sparsemax and Relaxed Wasserstein for Topic Sparsity](https://mlsanity.com/paper/1810.09079/) (2018)
- [Robust Estimation in Generalised Linear Models : The Density Power Divergence Approach](https://mlsanity.com/paper/1403.6606/) (2014)
- [A New Family of Divergences Originating from Model Adequacy Tests and Application to Robust Statistical Inference](https://mlsanity.com/paper/1611.05224/) (2016)
- [Robust Tests for the Equality of Two Normal Means based on the Density Power Divergence](https://mlsanity.com/paper/1403.1395/) (2014)
- [New improved estimators for overdispersion in models with clustered multinomial data and unequal cluster sizes](https://mlsanity.com/paper/1504.00397/) (2015)
- [Towards a better understanding of the dual representation of phi divergences](https://mlsanity.com/paper/1506.02166/) (2015)
- [Optimal transport with some directed distances](https://mlsanity.com/paper/2105.05989/) (2021)

## Abstract (arXiv)

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.