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In this paper we provide the asymptotic theory of the general of $\phi$-divergences measures, which include the most common divergence measures : Renyi and Tsallis families and the Kullback-Leibler measure. We are interested in divergence…

Statistics Theory · Mathematics 2019-04-01 Ba Amadou Diadie , Gane Samb Lo

We examine a family of intrinsic performance measures in terms of probability distributions that generalize Hellinger distance and Fisher information. They are applied to quantum metrology to assess the uncertainty in the detection of…

Quantum Physics · Physics 2015-03-20 Alfredo Luis , Alfonso Rodil

This paper describes a generalization of the Hellinger distance which we call the S -Hellinger distance; this general family connects the Hellinger distance smoothly with the $L_2$-divergence by a tuning parameter $\alpha$ and is indeed a…

Methodology · Statistics 2014-12-08 Abhik Ghosh , Ayanendranath Basu

We introduce generalized $(\alpha,\beta)$-transformations, which include all $(\alpha,\beta)$ and generalized $\beta$-transformations, and prove that all transitive generalized $(\alpha,\beta)$-transformations satisfy the level-2 large…

Dynamical Systems · Mathematics 2022-06-22 Mao Shinoda , Kenichiro Yamamoto

We introduce a new information-theoretic formulation of quantum measurement uncertainty relations, based on the notion of relative entropy between measurement probabilities. In the case of a finite-dimensional system and for any approximate…

Mathematical Physics · Physics 2018-03-02 Alberto Barchielli , Matteo Gregoratti , Alessandro Toigo

The weighted forms of generalized survival and failure entropies of order ($\alpha,\beta$) are proposed and some properties are obtained. We further propose the dynamic versions of weighted generalized survival and failures entropies and…

Statistics Theory · Mathematics 2020-06-15 Siddhartha Chakraborty , Biswabrata Pradhan

In this work, we study generalized entropies and information geometry in a group-theoretical framework. We explore the conditions that ensure the existence of some natural properties and at the same time of a group-theoretical structure for…

Mathematical Physics · Physics 2021-08-03 Miguel A. Rodríguez , Álvaro Romaniega , Piergiulio Tempesta

In any parametric inference problem, the robustness of the procedure is a real concern. A procedure which retains a high degree of efficiency under the model and simultaneously provides stable inference under data contamination is…

Methodology · Statistics 2020-01-01 Ayanendranath Basu , Abhijit Mandal , Nirian Martin , Leandro Pardo

The aim of this paper is to discuss new results concerning some kinds of parametric extended entropies and divergences. As a result of our studies for mathematical properties on entropy and divergence, we give new bounds for the Tsallis…

Information Theory · Computer Science 2019-07-24 Shigeru Furuichi , Nicuşor Minculete

The binary divergences that are divergences between probability measures defined on the same 2-point set have an interesting property. For the chi-squared divergence and the relative entropy, it is known that their binary divergence attain…

Information Theory · Computer Science 2021-02-09 Tomohiro Nishiyama

Entropy and relative or cross entropy measures are two very fundamental concepts in information theory and are also widely used for statistical inference across disciplines. The related optimization problems, in particular the maximization…

Statistics Theory · Mathematics 2021-06-18 Abhik Ghosh , Ayanendranath Basu

We prove characterization theorems for relative entropy (also known as Kullback-Leibler divergence), q-logarithmic entropy (also known as Tsallis entropy), and q-logarithmic relative entropy. All three have been characterized axiomatically…

Information Theory · Computer Science 2017-12-14 Tom Leinster

Quantum generalizations of Renyi's entropies are a useful tool to describe a variety of operational tasks in quantum information processing. Two families of such generalizations turn out to be particularly useful: the Petz quantum Renyi…

Quantum Physics · Physics 2017-01-25 Raban Iten , Joseph M. Renes , David Sutter

The most popular hypothesis testing procedure, the likelihood ratio test, is known to be highly non-robust in many real situations. Basu et al. (2013a) provided an alternative robust procedure of hypothesis testing based on the density…

Statistics Theory · Mathematics 2016-07-04 Abhik Ghosh , Ayanendranath Basu , Leandro Pardo

Minimization problems with respect to a one-parameter family of generalized relative entropies are studied. These relative entropies, which we term relative $\alpha$-entropies (denoted $\mathscr{I}_{\alpha}$), arise as redundancies under…

Information Theory · Computer Science 2015-06-11 M. Ashok Kumar , Rajesh Sundaresan

This paper generalizes beta divergence beyond its classical form associated with power variance functions of Tweedie models. Generalized form is represented by a compact definite integral as a function of variance function of the…

Machine Learning · Statistics 2013-06-19 Y. Kenan Yilmaz

We study minimization of a parametric family of relative entropies, termed relative $\alpha$-entropies (denoted $\mathscr{I}_{\alpha}(P,Q)$). These arise as redundancies under mismatched compression when cumulants of compressed lengths are…

Information Theory · Computer Science 2014-10-21 M. Ashok Kumar , Rajesh Sundaresan

The main purpose of this paper is to introduce and study the behavior of minimum {\phi}-divergence estimators as an alternative to the maximum likelihood estimator in latent class models for binary items. As it will become clear below,…

Methodology · Statistics 2014-06-03 Ángel Felipe , Pedro Miranda , Leandro Pardo

We introduce estimation and test procedures through divergence minimiza- tion for models satisfying linear constraints with unknown parameter. These procedures extend the empirical likelihood (EL) method and share common features with…

Statistics Theory · Mathematics 2016-11-25 Michel Broniatowski , Amor Keziou

We study an entropy measure for quantum systems that generalizes the von Neumann entropy as well as its classical counterpart, the Gibbs or Shannon entropy. The entropy measure is based on hypothesis testing and has an elegant formulation…

Quantum Physics · Physics 2014-02-19 F. Dupuis , L. Kraemer , P. Faist , J. M. Renes , R. Renner