arXiv subjects

Atin Gayen

Publications and source records attributed to Atin Gayen.

5 recordsLinked to original sources

arXiv2026

Quantum Relative-alpha-Entropies: A Structural and Geometric Perspective

Most quantum divergences derive their structure from classical f-divergences or Renyi-type constructions, a dependence that obscures several quantum geometric effects. We introduce a quantum relative-alpha-entropy that extends Umegaki's relative entropy while falling outside the f-divergence class. The proposed divergence exhibits a nonlinear convexity property, which yields a generalized convexity result for the Petz-Renyi divergence for alpha greater than one, complementing the known convexity for alpha less than one. It is additive under tensor products, invariant under unitary transformations, and depends only on the relative geometry of quantum states rather than their absolute magnitudes. Using Nussbaum-Szkola-type distributions, we also establish an exact correspondence of this divergence with classical relative-alpha-entropy. This reveals relative-alpha-entropy as a fundamentally geometric notion of quantum distinguishability not captured by existing divergence frameworks.

quant-ph↗

arXiv2024

Generalized Fisher-Darmois-Koopman-Pitman Theorem and Rao-Blackwell Type Estimators for Power-Law Distributions

This paper generalizes the notion of sufficiency for estimation problems beyond maximum likelihood. In particular, we consider estimation problems based on Jones et al. and Basu et al. likelihood functions that are popular among distance-based robust inference methods. We first characterize the probability distributions that always have a fixed number of sufficient statistics (independent of sample size) with respect to these likelihood functions. These distributions are power-law extensions of the usual exponential family and contain Student distributions as a special case. We then extend the notion of minimal sufficient statistics and compute it for these power-law families. Finally, we establish a Rao-Blackwell-type theorem for finding the best estimators for a power-law family. This helps us establish Cramér-Rao-type lower bounds for power-law families.

math.ST↗

arXiv2021

Projection Theorems and Estimating Equations for Power-Law Models

We extend projection theorems concerning Hellinger and Jones et al. divergences to the continuous case. These projection theorems reduce certain estimation problems on generalized exponential models to linear problems. We introduce the notion of regularity for generalized exponential models and show that the projection theorems in this case are similar to the ones in discrete and canonical case. We also apply these ideas to solve certain estimation problems concerning Student and Cauchy distributions.

math.ST↗

arXiv2018

Generalized Estimating Equation for the Student-t Distributions

In \cite{KumarS15J2}, it was shown that a generalized maximum likelihood estimation problem on a (canonical) $α$-power-law model ($\mathbb{M}^{(α)}$-family) can be solved by solving a system of linear equations. This was due to an orthogonality relationship between the $\mathbb{M}^{(α)}$-family and a linear family with respect to the relative $α$-entropy (or the $\mathscr{I}_α$-divergence). Relative $α$-entropy is a generalization of the usual relative entropy (or the Kullback-Leibler divergence). $\mathbb{M}^{(α)}$-family is a generalization of the usual exponential family. In this paper, we first generalize the $\mathbb{M}^{(α)}$-family including the multivariate, continuous case and show that the Student-t distributions fall in this family. We then extend the above stated result of \cite{KumarS15J2} to the general $\mathbb{M}^{(α)}$-family. Finally we apply this result to the Student-t distribution and find generalized estimators for its parameters.

math.ST↗

arXiv2017

Projection Theorems of Divergences and Likelihood Maximization Methods

Projection theorems of divergences enable us to find reverse projection of a divergence on a specific statistical model as a forward projection of the divergence on a different but rather "simpler" statistical model, which, in turn, results in solving a system of linear equations. Reverse projection of divergences are closely related to various estimation methods such as the maximum likelihood estimation or its variants in robust statistics. We consider projection theorems of three parametric families of divergences that are widely used in robust statistics, namely the Rényi divergences (or the Cressie-Reed power divergences), density power divergences, and the relative $α$-entropy (or the logarithmic density power divergences). We explore these projection theorems from the usual likelihood maximization approach and from the principle of sufficiency. In particular, we show the equivalence of solving the estimation problems by the projection theorems of the respective divergences and by directly solving the corresponding estimating equations. We also derive the projection theorem for the density power divergences.

cs.IT↗