A new class of metric divergences on probability spaces and its applicability in statistics

Österreicher, Ferdinand; Vajda, Igor

doi:10.1007/bf02517812

Cited by 180 publications

(141 citation statements)

References 22 publications

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“…Section 5 is devoted to the class of f -divergences of perimeter type, introduced and studied inÖsterreicher and Vajda (2003) and Vajda (2009). It is based on the class of entropies due to Arimoto (1971) and contains, next to the f -divergence given by (1), the total variation distance and a symmetrized version of the I-divergence also the squared Hellinger distance (with f (u) = ( √ u − 1) 2 ) and the squared Puri-Vincze distance.…”

Section: Fösterreichermentioning

confidence: 99%

Distances Based on the Perimeter of the Risk Set of a Testing Problem

Österreicher

2016

AJS

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At the core of this paper is a simple geometric object, namely the risk set of a statistical testing problem on the one hand and f -divergences, which were introduced by Csiszár (1963) on the other hand. f -divergences are measures for the hardness of a testing problem depending on a convex real valued function f on the interval [0, ∞). The choice of this parameter f can be adjusted so as to match the needs for specific applications.

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Section: Fösterreichermentioning

confidence: 99%

Distances Based on the Perimeter of the Risk Set of a Testing Problem

Österreicher

2016

AJS

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“…In effect, the use of this  in forming a convex combination of the two estimators can be viewed as pursuing an objective of minimizing the variation in the resultant estimator (12). If we make use of the optimum  in the optimal convex estimator in (12), the result comes out in the form of a Stein-like estimator [20][21], where for a given samples of data, shrinkage is from  …”

Section: Empirical Calculation Of mentioning

confidence: 99%

“…If we make use of the optimum  in the optimal convex estimator in (12), the result comes out in the form of a Stein-like estimator [20][21], where for a given samples of data, shrinkage is from  …”

Section: Empirical Calculation Of mentioning

confidence: 99%

“…In particular, as noted by [11][12], whether the statistic is designated as , the same collection of members of the family of divergence measures are ultimately spanned, when considering all of the possibilities for of the MPD estimators of obtained by optimizing the are consistent and asymptotically normally distributed. They are also asymptotically efficient, relative to the optimal estimating function (OptEF) estimator [14], when a uniform distribution, or equivalently the empirical distribution function (EDF), is used for the reference distribution.…”

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confidence: 99%

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Implications of the Cressie-Read Family of Additive Divergences for Information Recovery

Judge

Mittelhammer

2012

Entropy

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Abstract:To address the unknown nature of probability-sampling models, in this paper we use information theoretic concepts and the Cressie-Read (CR) family of information divergence measures to produce a flexible family of probability distributions, likelihood functions, estimators, and inference procedures. The usual case in statistical modeling is that the noisy indirect data are observed and known and the sampling model-error distribution-probability space, consistent with the data, is unknown. To address the unknown sampling process underlying the data, we consider a convex combination of two or more estimators derived from members of the flexible CR family of divergence measures and optimize that combination to select an estimator that minimizes expected quadratic loss. Sampling experiments are used to illustrate the finite sample properties of the resulting estimator and the nature of the recovered sampling distribution.

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“…Although Jensen-Shannon divergence does not guarantee the triangular inequality of a metric, the square root of the divergence follows the metric property (as shown in [27,28]). …”

Section: Description Of Js Divergencementioning

confidence: 99%