A Generalized Relative (α, β)-Entropy: Geometric Properties and Applications to Robust Statistical Inference

Entropy and relative entropy measures play a crucial role in mathematical information theory. The relative entropies are also widely used in statistics under the name of divergence measures which link these two fields of science through the minimum divergence principle. Divergence measures are popul...

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Vydáno v:Entropy (Basel, Switzerland) Ročník 20; číslo 5; s. 347
Hlavní autoři: Ghosh, Abhik, Basu, Ayanendranath
Médium: Journal Article
Jazyk:angličtina
Vydáno: Basel MDPI AG 06.05.2018
MDPI
Témata:
Convexity
Density
Divergence
Entropy
Entropy (Information theory)
generalized renyi entropy
Hypothesis testing
Information theory
logarithmic super divergence
minimum divergence inference
Outliers (statistics)
Parameter estimation
Parameter robustness
relative entropy
robustness
Robustness (mathematics)
Statistical inference
ISSN:1099-4300, 1099-4300
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Abstract Entropy and relative entropy measures play a crucial role in mathematical information theory. The relative entropies are also widely used in statistics under the name of divergence measures which link these two fields of science through the minimum divergence principle. Divergence measures are popular among statisticians as many of the corresponding minimum divergence methods lead to robust inference in the presence of outliers in the observed data; examples include the ϕ -divergence, the density power divergence, the logarithmic density power divergence and the recently developed family of logarithmic super divergence (LSD). In this paper, we will present an alternative information theoretic formulation of the LSD measures as a two-parameter generalization of the relative α -entropy, which we refer to as the general ( α , β ) -entropy. We explore its relation with various other entropies and divergences, which also generates a two-parameter extension of Renyi entropy measure as a by-product. This paper is primarily focused on the geometric properties of the relative ( α , β ) -entropy or the LSD measures; we prove their continuity and convexity in both the arguments along with an extended Pythagorean relation under a power-transformation of the domain space. We also derive a set of sufficient conditions under which the forward and the reverse projections of the relative ( α , β ) -entropy exist and are unique. Finally, we briefly discuss the potential applications of the relative ( α , β ) -entropy or the LSD measures in statistical inference, in particular, for robust parameter estimation and hypothesis testing. Our results on the reverse projection of the relative ( α , β ) -entropy establish, for the first time, the existence and uniqueness of the minimum LSD estimators. Numerical illustrations are also provided for the problem of estimating the binomial parameter.
AbstractList Entropy and relative entropy measures play a crucial role in mathematical information theory. The relative entropies are also widely used in statistics under the name of divergence measures which link these two fields of science through the minimum divergence principle. Divergence measures are popular among statisticians as many of the corresponding minimum divergence methods lead to robust inference in the presence of outliers in the observed data; examples include the ϕ -divergence, the density power divergence, the logarithmic density power divergence and the recently developed family of logarithmic super divergence (LSD). In this paper, we will present an alternative information theoretic formulation of the LSD measures as a two-parameter generalization of the relative α -entropy, which we refer to as the general ( α , β ) -entropy. We explore its relation with various other entropies and divergences, which also generates a two-parameter extension of Renyi entropy measure as a by-product. This paper is primarily focused on the geometric properties of the relative ( α , β ) -entropy or the LSD measures; we prove their continuity and convexity in both the arguments along with an extended Pythagorean relation under a power-transformation of the domain space. We also derive a set of sufficient conditions under which the forward and the reverse projections of the relative ( α , β ) -entropy exist and are unique. Finally, we briefly discuss the potential applications of the relative ( α , β ) -entropy or the LSD measures in statistical inference, in particular, for robust parameter estimation and hypothesis testing. Our results on the reverse projection of the relative ( α , β ) -entropy establish, for the first time, the existence and uniqueness of the minimum LSD estimators. Numerical illustrations are also provided for the problem of estimating the binomial parameter.
Entropy and relative entropy measures play a crucial role in mathematical information theory. The relative entropies are also widely used in statistics under the name of divergence measures which link these two fields of science through the minimum divergence principle. Divergence measures are popular among statisticians as many of the corresponding minimum divergence methods lead to robust inference in the presence of outliers in the observed data; examples include the ϕ -divergence, the density power divergence, the logarithmic density power divergence and the recently developed family of logarithmic super divergence (LSD). In this paper, we will present an alternative information theoretic formulation of the LSD measures as a two-parameter generalization of the relative α -entropy, which we refer to as the general ( α , β ) -entropy. We explore its relation with various other entropies and divergences, which also generates a two-parameter extension of Renyi entropy measure as a by-product. This paper is primarily focused on the geometric properties of the relative ( α , β ) -entropy or the LSD measures; we prove their continuity and convexity in both the arguments along with an extended Pythagorean relation under a power-transformation of the domain space. We also derive a set of sufficient conditions under which the forward and the reverse projections of the relative ( α , β ) -entropy exist and are unique. Finally, we briefly discuss the potential applications of the relative ( α , β ) -entropy or the LSD measures in statistical inference, in particular, for robust parameter estimation and hypothesis testing. Our results on the reverse projection of the relative ( α , β ) -entropy establish, for the first time, the existence and uniqueness of the minimum LSD estimators. Numerical illustrations are also provided for the problem of estimating the binomial parameter.Entropy and relative entropy measures play a crucial role in mathematical information theory. The relative entropies are also widely used in statistics under the name of divergence measures which link these two fields of science through the minimum divergence principle. Divergence measures are popular among statisticians as many of the corresponding minimum divergence methods lead to robust inference in the presence of outliers in the observed data; examples include the ϕ -divergence, the density power divergence, the logarithmic density power divergence and the recently developed family of logarithmic super divergence (LSD). In this paper, we will present an alternative information theoretic formulation of the LSD measures as a two-parameter generalization of the relative α -entropy, which we refer to as the general ( α , β ) -entropy. We explore its relation with various other entropies and divergences, which also generates a two-parameter extension of Renyi entropy measure as a by-product. This paper is primarily focused on the geometric properties of the relative ( α , β ) -entropy or the LSD measures; we prove their continuity and convexity in both the arguments along with an extended Pythagorean relation under a power-transformation of the domain space. We also derive a set of sufficient conditions under which the forward and the reverse projections of the relative ( α , β ) -entropy exist and are unique. Finally, we briefly discuss the potential applications of the relative ( α , β ) -entropy or the LSD measures in statistical inference, in particular, for robust parameter estimation and hypothesis testing. Our results on the reverse projection of the relative ( α , β ) -entropy establish, for the first time, the existence and uniqueness of the minimum LSD estimators. Numerical illustrations are also provided for the problem of estimating the binomial parameter.
Author Basu, Ayanendranath
Ghosh, Abhik
AuthorAffiliation Indian Statistical Institute, Kolkata 700108, India
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Cites_doi 10.1103/PhysRevLett.102.063001
10.1109/18.179378
10.1080/02331880902986919
10.1016/j.jmva.2012.10.003
10.1103/PhysRevE.97.042107
10.1103/PhysRevE.90.052149
10.1214/aoms/1177728353
10.1007/BF00773476
10.1109/LSP.2016.2606661
10.1111/j.1538-4632.2010.00800.x
10.1007/s00145-017-9265-9
10.1016/j.physa.2010.03.045
10.3390/e17075043
10.1214/aoms/1177729487
10.1109/TIT.2015.2449312
10.1103/PhysRevLett.105.022002
10.1109/TIT.2015.2449311
10.1016/j.neucom.2013.12.018
10.1007/s41884-018-0002-8
10.1111/j.2517-6161.1984.tb01318.x
10.1007/JHEP01(2018)115
10.1016/j.dsp.2017.06.019
10.1214/aoms/1177728660
10.1201/b19113
10.1103/PhysRevD.83.052004
10.1109/TSP.2015.2468677
10.1287/moor.1100.0449
10.1002/j.1538-7305.1948.tb01338.x
10.1063/1.1721117
10.1201/b10956
10.1016/j.physa.2005.06.065
10.1109/TASLP.2015.2427520
10.1109/TIT.1981.1056374
10.1109/JRPROC.1949.232969
10.1103/PhysRevLett.102.097202
10.1080/03610918.2017.1366514
10.1007/s00477-016-1221-y
10.1088/1742-5468/2011/03/P03008
10.1103/PhysRevLett.96.110601
10.1016/j.dam.2014.10.004
10.1103/PhysRevLett.111.180604
10.1016/j.patcog.2016.07.038
10.1063/1.4994922
10.3390/e14091606
10.1137/140962802
10.1109/TIE.2014.2370936
10.1007/s11634-010-0061-8
10.1214/aoms/1177729694
10.1016/j.physa.2017.09.020
10.1162/neco.2008.04-08-771
10.1155/2014/505184
10.1016/j.jmva.2008.02.004
10.1109/ISIT.2016.7541474
10.1103/PhysRevLett.100.055003
10.3390/e19120664
10.1109/TR.2016.2570549
10.1016/j.engappai.2015.05.004
10.1016/S0019-9958(65)90332-3
10.1109/TIT.2006.881731
10.1561/9781933019543
10.1109/TIT.2006.887466
10.1063/1.2423258
10.1093/biomet/88.3.865
10.1007/s00161-004-0174-4
10.1103/PhysRevA.67.051402
10.1007/JHEP08(2011)086
10.1109/TIT.2004.840871
10.1016/j.physletb.2017.10.043
10.1007/978-94-009-6581-2
10.2174/2213275910666170307111644
10.1111/j.2517-6161.1995.tb02050.x
10.1109/TIT.1980.1056144
10.1214/aos/1176325512
10.1007/BF02018661
10.1109/18.481781
10.1007/BF01016429
10.1109/TIT.1956.1056823
10.1109/TIT.2011.2159046
10.1016/S0167-8140(15)30456-4
10.1016/j.spl.2008.04.011
10.1007/s10182-015-0252-x
10.1038/ncomms12472
10.1016/j.jmva.2008.03.011
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References Devoe (ref_86) 2009; 102
Lutwak (ref_72) 2005; 51
Jones (ref_74) 2001; 88
Teboulle (ref_107) 1993; 39
ref_14
ref_96
Toma (ref_113) 2010; 115
Arikan (ref_44) 1996; 42
Presse (ref_94) 2014; 90
Marazzi (ref_112) 2010; 4
Pickup (ref_85) 2009; 102
Stummer (ref_70) 2010; 44
Jamaati (ref_65) 2018; 490
Nielsen (ref_24) 2016; 23
Kolmogorov (ref_8) 1956; IT-2
Tusnady (ref_21) 1984; 1
Kullback (ref_13) 1951; 22
ref_22
ref_20
Liu (ref_84) 2008; 100
(ref_57) 1978; 39
Sundaresan (ref_17) 2007; 53
Shannon (ref_2) 1949; 37
(ref_18) 1975; 3
Cressie (ref_97) 1984; 46
ref_71
Kullback (ref_11) 1954; 25
Maji (ref_79) 2016; 100
Kumar (ref_73) 2015; 61
ref_78
Fevotte (ref_106) 2009; 21
Pinski (ref_28) 2015; 47
Maji (ref_80) 2017; 2
Fujisawa (ref_76) 2013; 7
Majhi (ref_90) 2017; 775
Ferracuti (ref_42) 2015; 44
Windham (ref_75) 1995; 57
Lutz (ref_81) 2003; 67
(ref_102) 1991; 19
Broniatowski (ref_109) 2009; 100
Tsallis (ref_61) 2004; 16
Khinchin (ref_5) 1956; 11
Wei (ref_47) 2018; 97
Monthus (ref_31) 2011; 2011
Shore (ref_91) 1980; 26
Costa (ref_23) 2015; 197
Attouch (ref_29) 2010; 35
Eliazar (ref_30) 2010; 389
Giantomassi (ref_39) 2015; 62
Dong (ref_52) 2016; 7
Leise (ref_67) 2006; 52
Kumar (ref_59) 2013; 9
Khinchin (ref_4) 1953; 8
Caticha (ref_92) 2006; 872
Kullback (ref_10) 1953; 24
Bafroui (ref_32) 2014; 133
Shannon (ref_1) 1948; 27
Kusuki (ref_53) 2018; 2018
Sriraman (ref_36) 2017; 147
Markel (ref_50) 2014; 111
Kumbhakar (ref_54) 2017; 31
Kullback (ref_12) 1956; 27
Lindsay (ref_103) 1994; 22
Toma (ref_111) 2009; 79
Kullback (ref_9) 1952; 23
Vo (ref_38) 2015; 63
Campbell (ref_45) 1965; 8
Nie (ref_56) 2017; 10
ref_69
ref_68
(ref_100) 1967; 2
ref_66
Sun (ref_37) 2015; 23
Harmouche (ref_40) 2016; 65
ref_64
ref_63
ref_62
Nielsen (ref_27) 2011; 57
Kumar (ref_60) 2014; 2014
Basu (ref_108) 1994; 46
Esteban (ref_104) 1995; 31
Amari (ref_25) 2010; 58
ref_35
Khachatryan (ref_87) 2010; 105
Chatrchyan (ref_88) 2011; 2011
ref_34
Douglas (ref_82) 2006; 96
Cover (ref_15) 1981; 27
Adare (ref_89) 2011; 83
Presse (ref_95) 2015; 17
(ref_26) 2012; 14
Bai (ref_51) 2018; 31
(ref_101) 1972; 2
(ref_98) 1963; 3
Broniatowski (ref_110) 2012; 48
ref_105
ref_46
Fujisawa (ref_77) 2008; 99
Hua (ref_41) 2017; 69
ref_3
(ref_19) 1984; 12
(ref_99) 1967; 2
Xing (ref_55) 2017; 61
Batty (ref_33) 2010; 42
ref_49
ref_48
Presse (ref_93) 2013; 111
Matthews (ref_43) 2016; 51
Burlaga (ref_83) 2005; 356
Kumar (ref_16) 2015; 61
ref_7
ref_6
Tsallis (ref_58) 1988; 52
References_xml – volume: 102
  start-page: 063001
  year: 2009
  ident: ref_86
  article-title: Power-Law Distributions for a Trapped Ion Interacting with a Classical Buffer Gas
  publication-title: Phys. Rev. Lett.
  doi: 10.1103/PhysRevLett.102.063001
– volume: 3
  start-page: 146
  year: 1975
  ident: ref_18
  article-title: I-divergence geometry of probability distributions and minimization problems
  publication-title: Ann. Probab.
– volume: 2
  start-page: 299
  year: 1967
  ident: ref_99
  article-title: Information-type measures of difference of probability distributions and indirect observations
  publication-title: Stud. Scientiarum Math. Hung.
– volume: 39
  start-page: 297
  year: 1993
  ident: ref_107
  article-title: Convergence of best ϕ-entropy estimates
  publication-title: IEEE Trans. Inf. Theory
  doi: 10.1109/18.179378
– volume: 44
  start-page: 169
  year: 2010
  ident: ref_70
  article-title: On divergences of finite measures and their applicability in statistics and information theory
  publication-title: Statistics
  doi: 10.1080/02331880902986919
– ident: ref_68
– volume: 115
  start-page: 359
  year: 2010
  ident: ref_113
  article-title: Optimal robust M-estimators using Renyi pseudodistances
  publication-title: J. Multivar. Anal.
  doi: 10.1016/j.jmva.2012.10.003
– volume: 97
  start-page: 042107
  year: 2018
  ident: ref_47
  article-title: Relations between heat exchange and Rényi divergences
  publication-title: Phys. Rev. E
  doi: 10.1103/PhysRevE.97.042107
– volume: 90
  start-page: 052149
  year: 2014
  ident: ref_94
  article-title: Nonadditive entropy maximization is inconsistent with Bayesian updating
  publication-title: Phys. Rev. E
  doi: 10.1103/PhysRevE.90.052149
– volume: 51
  start-page: 231
  year: 2016
  ident: ref_43
  article-title: On sparse variational methods and the Kullback-Leibler divergence between stochastic processes
  publication-title: J. Mach. Learn. Res.
– volume: 27
  start-page: 122
  year: 1956
  ident: ref_12
  article-title: An application of information theory to multivariate analysis II
  publication-title: Ann. Math. Stat.
  doi: 10.1214/aoms/1177728353
– volume: 46
  start-page: 683
  year: 1994
  ident: ref_108
  article-title: Minimum disparity estimation for continuous models: Efficiency, distributions and robustness
  publication-title: Ann. Inst. Stat. Math.
  doi: 10.1007/BF00773476
– ident: ref_71
– volume: 23
  start-page: 1543
  year: 2016
  ident: ref_24
  article-title: Guaranteed bounds on the Kullback-Leibler divergence of univariate mixtures
  publication-title: IEEE Signal Process. Lett.
  doi: 10.1109/LSP.2016.2606661
– volume: 42
  start-page: 395
  year: 2010
  ident: ref_33
  article-title: Space, Scale, and Scaling in Entropy Maximizing
  publication-title: Geogr. Anal.
  doi: 10.1111/j.1538-4632.2010.00800.x
– volume: 31
  start-page: 610
  year: 2018
  ident: ref_51
  article-title: Improved security proofs in lattice-based cryptography: Using the Rényi divergence rather than the statistical distance
  publication-title: J. Cryptol.
  doi: 10.1007/s00145-017-9265-9
– volume: 389
  start-page: 3023
  year: 2010
  ident: ref_30
  article-title: Maximization of statistical heterogeneity: From Shannon’s entropy to Gini’s index
  publication-title: Phys. A Stat. Mech. Appl.
  doi: 10.1016/j.physa.2010.03.045
– volume: 17
  start-page: 5043
  year: 2015
  ident: ref_95
  article-title: Reply to C. Tsallis’ “Conceptual Inadequacy of the Shore and Johnson Axioms for Wide Classes of Complex Systems”
  publication-title: Entropy
  doi: 10.3390/e17075043
– volume: 19
  start-page: 2032
  year: 1991
  ident: ref_102
  article-title: Why least squares and maximum entropy? An axiomatic approach to inference for linear inverse problems
  publication-title: Ann. Stat.
– volume: 23
  start-page: 88
  year: 1952
  ident: ref_9
  article-title: An application of information theory to multivariate analysis
  publication-title: Ann. Math. Stat.
  doi: 10.1214/aoms/1177729487
– volume: 61
  start-page: 5081
  year: 2015
  ident: ref_73
  article-title: Minimization Problems Based on Relative α-Entropy II: Reverse Projection
  publication-title: IEEE Trans. Infor. Theory
  doi: 10.1109/TIT.2015.2449312
– volume: 105
  start-page: 022002
  year: 2010
  ident: ref_87
  article-title: Transverse-Momentum and Pseudorapidity Distributions of Charged Hadrons in pp Collisions at s = 7 TeV
  publication-title: Phys. Rev. Lett.
  doi: 10.1103/PhysRevLett.105.022002
– volume: 61
  start-page: 5063
  year: 2015
  ident: ref_16
  article-title: Minimization Problems Based on Relative α-Entropy I: Forward Projection
  publication-title: IEEE Trans. Inf. Theory
  doi: 10.1109/TIT.2015.2449311
– volume: 133
  start-page: 437
  year: 2014
  ident: ref_32
  article-title: Application of wavelet energy and Shannon entropy for feature extraction in gearbox fault detection under varying speed conditions
  publication-title: Neurocomputing
  doi: 10.1016/j.neucom.2013.12.018
– ident: ref_22
  doi: 10.1007/s41884-018-0002-8
– volume: 46
  start-page: 440
  year: 1984
  ident: ref_97
  article-title: Multinomial goodness-of-fit tests
  publication-title: J. R. Stat. Soc. B
  doi: 10.1111/j.2517-6161.1984.tb01318.x
– volume: 2018
  start-page: 115
  year: 2018
  ident: ref_53
  article-title: Renyi entropy for local quenches in 2D CFT from numerical conformal blocks
  publication-title: J. High Energy Phys.
  doi: 10.1007/JHEP01(2018)115
– ident: ref_62
– volume: 69
  start-page: 106
  year: 2017
  ident: ref_41
  article-title: Matrix CFAR detectors based on symmetrized Kullback-Leibler and total Kullback-Leibler divergences
  publication-title: Digit. Signal Process.
  doi: 10.1016/j.dsp.2017.06.019
– volume: 31
  start-page: 337
  year: 1995
  ident: ref_104
  article-title: A summary of entropy statistics
  publication-title: Kybernetica
– volume: 25
  start-page: 745
  year: 1954
  ident: ref_11
  article-title: Certain inequalities in information theory and the Cramer-Rao inequality
  publication-title: Ann. Math. Stat.
  doi: 10.1214/aoms/1177728660
– ident: ref_63
  doi: 10.1201/b19113
– volume: 83
  start-page: 052004
  year: 2011
  ident: ref_89
  article-title: Measurement of neutral mesons in p + p collisions at s = 200 GeV and scaling properties of hadron production
  publication-title: Phys. Rev. D
  doi: 10.1103/PhysRevD.83.052004
– ident: ref_3
– volume: 63
  start-page: 5812
  year: 2015
  ident: ref_38
  article-title: Derivation of the PHD and CPHD Filters Based on Direct Kullback-Leibler Divergence Minimization
  publication-title: IEEE Trans. Signal Process.
  doi: 10.1109/TSP.2015.2468677
– volume: 35
  start-page: 438
  year: 2010
  ident: ref_29
  article-title: Proximal alternating minimization and projection methods for nonconvex problems: An approach based on the Kurdyka-Lojasiewicz inequality
  publication-title: Math. Oper. Res.
  doi: 10.1287/moor.1100.0449
– volume: 27
  start-page: 379
  year: 1948
  ident: ref_1
  article-title: A mathematical theory of communication
  publication-title: Bell Syst. Tech. J.
  doi: 10.1002/j.1538-7305.1948.tb01338.x
– volume: 24
  start-page: 106
  year: 1953
  ident: ref_10
  article-title: A note on information theory
  publication-title: J. Appl. Phys.
  doi: 10.1063/1.1721117
– ident: ref_66
  doi: 10.1201/b10956
– volume: 356
  start-page: 375
  year: 2005
  ident: ref_83
  article-title: Triangle for the entropic index q of non-extensive statistical mechanics observed by Voyager 1 in the distant heliosphere
  publication-title: Phys. A Stat. Mech. Appl.
  doi: 10.1016/j.physa.2005.06.065
– volume: 23
  start-page: 1233
  year: 2015
  ident: ref_37
  article-title: Speech enhancement under low SNR conditions via noise estimation using sparse and low-rank NMF with Kullback-Leibler divergence
  publication-title: IEEE Trans. Audio Speech Lang. Process.
  doi: 10.1109/TASLP.2015.2427520
– volume: 27
  start-page: 483
  year: 1981
  ident: ref_15
  article-title: Maximum entropy and conditional probability
  publication-title: IEEE Trans. Inf. Theory
  doi: 10.1109/TIT.1981.1056374
– ident: ref_6
– volume: 37
  start-page: 10
  year: 1949
  ident: ref_2
  article-title: Communication in the presence of noise
  publication-title: Proc. IRE
  doi: 10.1109/JRPROC.1949.232969
– volume: 102
  start-page: 097202
  year: 2009
  ident: ref_85
  article-title: Generalized Spin-Glass Relaxation
  publication-title: Phys. Rev. Lett.
  doi: 10.1103/PhysRevLett.102.097202
– volume: 11
  start-page: 17
  year: 1956
  ident: ref_5
  article-title: On the fundamental theorems of information theory
  publication-title: Uspekhi Matematicheskikh Nauk
– ident: ref_49
  doi: 10.1080/03610918.2017.1366514
– volume: 31
  start-page: 949
  year: 2017
  ident: ref_54
  article-title: One-Dimensional velocity distribution in open channels using Renyi entropy
  publication-title: Stoch. Environ. Res. Risk Assess.
  doi: 10.1007/s00477-016-1221-y
– volume: 2011
  start-page: P03008
  year: 2011
  ident: ref_31
  article-title: Non-equilibrium steady states: Maximization of the Shannon entropy associated with the distribution of dynamical trajectories in the presence of constraints
  publication-title: J. Stat. Mech. Theory Exp.
  doi: 10.1088/1742-5468/2011/03/P03008
– ident: ref_64
– volume: 96
  start-page: 110601
  year: 2006
  ident: ref_82
  article-title: Tunable Tsallis Distributions in Dissipative Optical Lattices
  publication-title: Phys. Rev. Lett.
  doi: 10.1103/PhysRevLett.96.110601
– volume: 197
  start-page: 59
  year: 2015
  ident: ref_23
  article-title: Fisher information distance: A geometrical reading
  publication-title: Discret. Appl. Math.
  doi: 10.1016/j.dam.2014.10.004
– volume: 111
  start-page: 180604
  year: 2013
  ident: ref_93
  article-title: Nonadditive Entropies Yield Probability Distributions with Biases not Warranted by the Data
  publication-title: Phys. Rev. Lett.
  doi: 10.1103/PhysRevLett.111.180604
– volume: 61
  start-page: 185
  year: 2017
  ident: ref_55
  article-title: Selective ensemble of SVDDs with Renyi entropy based diversity measure
  publication-title: Pattern Recog.
  doi: 10.1016/j.patcog.2016.07.038
– volume: 147
  start-page: 044304
  year: 2017
  ident: ref_36
  article-title: Characteristic features of the Shannon information entropy of dipolar Bose-Einstein condensates
  publication-title: J. Chem. Phys.
  doi: 10.1063/1.4994922
– ident: ref_78
– volume: 14
  start-page: 1606
  year: 2012
  ident: ref_26
  article-title: Kullback-Leibler divergence measure for multivariate skew-normal distributions
  publication-title: Entropy
  doi: 10.3390/e14091606
– volume: 47
  start-page: 4091
  year: 2015
  ident: ref_28
  article-title: Kullback–Leibler approximation for probability measures on infinite dimensional spaces
  publication-title: SIAM J. Math. Anal.
  doi: 10.1137/140962802
– volume: 62
  start-page: 1770
  year: 2015
  ident: ref_39
  article-title: Electric motor fault detection and diagnosis by kernel density estimation and Kullback-Leibler divergence based on stator current measurements
  publication-title: IEEE Trans. Ind. Electron.
  doi: 10.1109/TIE.2014.2370936
– volume: 4
  start-page: 169
  year: 2010
  ident: ref_112
  article-title: Optimal robust estimates using the Hellinger distance
  publication-title: Adv. Data Anal. Classif.
  doi: 10.1007/s11634-010-0061-8
– volume: 22
  start-page: 79
  year: 1951
  ident: ref_13
  article-title: On information and sufficiency
  publication-title: Ann. Math. Stat.
  doi: 10.1214/aoms/1177729694
– volume: 490
  start-page: 1368
  year: 2018
  ident: ref_65
  article-title: Text mining by Tsallis entropy
  publication-title: Phys. A Stat. Mech. Appl.
  doi: 10.1016/j.physa.2017.09.020
– volume: 21
  start-page: 793
  year: 2009
  ident: ref_106
  article-title: Nonnegative Matrix Factorization with the Itakura–Saito Divergence: With application to music analysis
  publication-title: Neural Comput.
  doi: 10.1162/neco.2008.04-08-771
– volume: 2014
  start-page: 505184
  year: 2014
  ident: ref_60
  article-title: A generalization of the Havrda-Charvat and Tsallis entropy and its axiomatic characterization
  publication-title: Abstr. Appl. Anal.
  doi: 10.1155/2014/505184
– volume: 99
  start-page: 2053
  year: 2008
  ident: ref_77
  article-title: Robust parameter estimation with a small bias against heavy contamination
  publication-title: J. Multivar. Anal.
  doi: 10.1016/j.jmva.2008.02.004
– ident: ref_35
– ident: ref_48
  doi: 10.1109/ISIT.2016.7541474
– volume: 100
  start-page: 055003
  year: 2008
  ident: ref_84
  article-title: Superdiffusion and Non-Gaussian Statistics in a Driven-Dissipative 2D Dusty Plasma
  publication-title: Phys. Rev. Lett.
  doi: 10.1103/PhysRevLett.100.055003
– ident: ref_96
  doi: 10.3390/e19120664
– volume: 1
  start-page: 205
  year: 1984
  ident: ref_21
  article-title: Information geometry and alternating minimization procedures
  publication-title: Stat. Decis.
– volume: 65
  start-page: 1360
  year: 2016
  ident: ref_40
  article-title: Statistical approach for nondestructive incipient crack detection and characterization using Kullback-Leibler divergence
  publication-title: IEEE Trans. Reliab.
  doi: 10.1109/TR.2016.2570549
– volume: 44
  start-page: 25
  year: 2015
  ident: ref_42
  article-title: Electric motor defects diagnosis based on kernel density estimation and Kullback-Leibler divergence in quality control scenario
  publication-title: Eng. Appl. Artif. Intell.
  doi: 10.1016/j.engappai.2015.05.004
– volume: 3
  start-page: 85
  year: 1963
  ident: ref_98
  article-title: Eine informations theoretische Ungleichung und ihre Anwendung auf den Beweis der Ergodizitat von Markoffschen Ketten
  publication-title: Publ. Math. Inst. Hung. Acad. Sci.
– ident: ref_69
– volume: 8
  start-page: 423
  year: 1965
  ident: ref_45
  article-title: A coding theorem and Renyi’s entropy
  publication-title: Inf. Control
  doi: 10.1016/S0019-9958(65)90332-3
– volume: 52
  start-page: 4394
  year: 2006
  ident: ref_67
  article-title: On divergence and information in statistics and information theory
  publication-title: IEEE Trans. Inf. Theory
  doi: 10.1109/TIT.2006.881731
– volume: 7
  start-page: 1587
  year: 2013
  ident: ref_76
  article-title: Normalized estimating equation for robust parameter estimation
  publication-title: Elect. J. Stat.
– ident: ref_20
  doi: 10.1561/9781933019543
– volume: 53
  start-page: 269
  year: 2007
  ident: ref_17
  article-title: Guessing under source uncertainty
  publication-title: Proc. IEEE Trans. Inf. Theory
  doi: 10.1109/TIT.2006.887466
– ident: ref_7
– volume: 872
  start-page: 31
  year: 2006
  ident: ref_92
  article-title: Updating Probabilities
  publication-title: AIP Conf. Proc.
  doi: 10.1063/1.2423258
– volume: 88
  start-page: 865
  year: 2001
  ident: ref_74
  article-title: A comparison of related density-based minimum divergence estimators
  publication-title: Biometrika
  doi: 10.1093/biomet/88.3.865
– volume: 16
  start-page: 223
  year: 2004
  ident: ref_61
  article-title: Nonextensive statistical mechanics: A brief introduction
  publication-title: Contin. Mech. Thermodyn.
  doi: 10.1007/s00161-004-0174-4
– volume: 48
  start-page: 600
  year: 2012
  ident: ref_110
  article-title: Several applications of divergence criteria in continuous families
  publication-title: Kybernetika
– volume: 67
  start-page: 051402
  year: 2003
  ident: ref_81
  article-title: Anomalous diffusion and Tsallis statistics in an optical lattice
  publication-title: Phys. Rev. A
  doi: 10.1103/PhysRevA.67.051402
– ident: ref_34
– volume: 2011
  start-page: 86
  year: 2011
  ident: ref_88
  article-title: Charged particle transverse momentum spectra in pp collisions at s = 0.9 and 7 TeV
  publication-title: J. High Energy Phys.
  doi: 10.1007/JHEP08(2011)086
– volume: 51
  start-page: 473
  year: 2005
  ident: ref_72
  article-title: Cramear-Rao and moment-entropy inequalities for Renyi entropy and generalized Fisher information
  publication-title: IEEE Trans. Inf. Theory
  doi: 10.1109/TIT.2004.840871
– volume: 775
  start-page: 32
  year: 2017
  ident: ref_90
  article-title: Non-extensive statistical mechanics and black hole entropy from quantum geometry
  publication-title: Phys. Lett. B
  doi: 10.1016/j.physletb.2017.10.043
– ident: ref_14
  doi: 10.1007/978-94-009-6581-2
– volume: 10
  start-page: 122
  year: 2017
  ident: ref_56
  article-title: An Image Segmentation Method Based on Renyi Relative Entropy and Gaussian Distribution
  publication-title: Recent Patents Comput. Sci.
  doi: 10.2174/2213275910666170307111644
– volume: 57
  start-page: 599
  year: 1995
  ident: ref_75
  article-title: Robustifying model fitting
  publication-title: J. R. Stat. Soc. Ser. B
  doi: 10.1111/j.2517-6161.1995.tb02050.x
– volume: 26
  start-page: 26
  year: 1980
  ident: ref_91
  article-title: Axiomatic Derivation of the Principle of Maximum Entropy and the Principle of Minimum Cross-Entropy
  publication-title: IEEE Trans. Inf. Theory
  doi: 10.1109/TIT.1980.1056144
– volume: 2
  start-page: 329
  year: 1967
  ident: ref_100
  article-title: On topological properties of f-divergences
  publication-title: Stud. Scientiarum Math. Hung.
– volume: 58
  start-page: 183
  year: 2010
  ident: ref_25
  article-title: Information geometry of divergence functions
  publication-title: Bull. Pol. Acad. Sci. Tech. Sci.
– volume: 22
  start-page: 1081
  year: 1994
  ident: ref_103
  article-title: Efficiency versus robustness: The case for minimum Hellinger distance and related methods
  publication-title: Ann. Stat.
  doi: 10.1214/aos/1176325512
– volume: 9
  start-page: 95
  year: 2013
  ident: ref_59
  article-title: Axioms for (α, β, γ)-entropy of a generalized probability scheme
  publication-title: J. Appl. Math. Stat. Inf.
– volume: 2
  start-page: 191
  year: 1972
  ident: ref_101
  article-title: A class of measures of informativity of observation channels
  publication-title: Priodica Math. Hung.
  doi: 10.1007/BF02018661
– volume: 42
  start-page: 99
  year: 1996
  ident: ref_44
  article-title: An inequality on guessing and its application to sequential decoding
  publication-title: IEEE Trans. Inf. Theory
  doi: 10.1109/18.481781
– volume: 12
  start-page: 768
  year: 1984
  ident: ref_19
  article-title: Sanov property, generalized I -projection, and a conditional limit theorem
  publication-title: Ann. Probab.
– volume: 52
  start-page: 479
  year: 1988
  ident: ref_58
  article-title: Possible generalization of Boltzmann-Gibbs statistics
  publication-title: J. Stat. Phys
  doi: 10.1007/BF01016429
– volume: IT-2
  start-page: 102
  year: 1956
  ident: ref_8
  article-title: On the Shannon theory of information transmission in the case of continuous signals
  publication-title: IRE Trans. Inf. Theory
  doi: 10.1109/TIT.1956.1056823
– volume: 57
  start-page: 5455
  year: 2011
  ident: ref_27
  article-title: The Burbea-Rao and Bhattacharyya Centroids
  publication-title: IEEE Trans. Inf. Theory
  doi: 10.1109/TIT.2011.2159046
– ident: ref_46
– volume: 111
  start-page: S134
  year: 2014
  ident: ref_50
  article-title: PD-0351: Development of a novel regmentation framework using the Jensen Renyi divergence for adaptive radiotherapy
  publication-title: Radiother. Oncol.
  doi: 10.1016/S0167-8140(15)30456-4
– volume: 79
  start-page: 1
  year: 2009
  ident: ref_111
  article-title: Optimal robust M-estimators using divergences
  publication-title: Stat. Probab. Lett.
  doi: 10.1016/j.spl.2008.04.011
– volume: 100
  start-page: 99
  year: 2016
  ident: ref_79
  article-title: The Logarithmic Super Divergence and Asymptotic Inference Properties
  publication-title: AStA Adv. Stat. Anal.
  doi: 10.1007/s10182-015-0252-x
– ident: ref_105
– volume: 8
  start-page: 3
  year: 1953
  ident: ref_4
  article-title: The entropy concept in probability theory
  publication-title: Uspekhi Matematicheskikh Nauk
– volume: 7
  start-page: 12472
  year: 2016
  ident: ref_52
  article-title: The gravity dual of Rényi entropy
  publication-title: Nat. Commun.
  doi: 10.1038/ncomms12472
– volume: 100
  start-page: 16
  year: 2009
  ident: ref_109
  article-title: Parametric estimation and tests through divergences and the duality technique
  publication-title: J. Multivar. Anal.
  doi: 10.1016/j.jmva.2008.03.011
– volume: 2
  start-page: 39
  year: 2017
  ident: ref_80
  article-title: Statistical Inference Based on the Logarithmic Power Divergence
  publication-title: Rashi
– volume: 39
  start-page: 277
  year: 1978
  ident: ref_57
  article-title: f-entropies, probability of error, and feature selection
  publication-title: Inf. Control
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Snippet Entropy and relative entropy measures play a crucial role in mathematical information theory. The relative entropies are also widely used in statistics under...
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SubjectTerms Convexity
Density
Divergence
Entropy
Entropy (Information theory)
generalized renyi entropy
Hypothesis testing
Information theory
logarithmic super divergence
minimum divergence inference
Outliers (statistics)
Parameter estimation
Parameter robustness
relative entropy
robustness
Robustness (mathematics)
Statistical inference
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