Estimating cell frequencies under inequality constraints based on the Kullback-Leibler information

This article considers the problem of estimating the cell frequencies in a contingency table under inequality constraints. Algorithms are proposed for cell frequency estimation via minimizing the Kullback–Leibler distance subject to inequality constraints. The proposed algorithms are shown to be sim...

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Published in:Statistica Neerlandica Vol. 66; no. 2; pp. 203 - 216
Main Authors: Fu, Lianyan, Gao, Wei, Shi, Ning-Zhong
Format: Journal Article
Language:English
Published: Oxford, UK Blackwell Publishing Ltd 01.05.2012
Netherlands Society for Statistics and Operations Research
Series:Statistica Neerlandica
Subjects:
ISSN:0039-0402, 1467-9574
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Abstract This article considers the problem of estimating the cell frequencies in a contingency table under inequality constraints. Algorithms are proposed for cell frequency estimation via minimizing the Kullback–Leibler distance subject to inequality constraints. The proposed algorithms are shown to be simple, easy to be used, fast, and reliable. Theorems are derived to guarantee the convergence of the algorithms. Applications and extensions of the algorithms are provided for more general problems than contingency table. The R programs that implement the proposed algorithms are presented in Appendix B.
AbstractList This article considers the problem of estimating the cell frequencies in a contingency table under inequality constraints. Algorithms are proposed for cell frequency estimation via minimizing the Kullback–Leibler distance subject to inequality constraints. The proposed algorithms are shown to be simple, easy to be used, fast, and reliable. Theorems are derived to guarantee the convergence of the algorithms. Applications and extensions of the algorithms are provided for more general problems than contingency table. The R programs that implement the proposed algorithms are presented in Appendix B.
This article considers the problem of estimating the cell frequencies in a contingency table under inequality constraints. Algorithms are proposed for cell frequency estimation via minimizing the Kullback-Leibler distance subject to inequality constraints. The proposed algorithms are shown to be simple, easy to be used, fast, and reliable. Theorems are derived to guarantee the convergence of the algorithms. Applications and extensions of the algorithms are provided for more general problems than contingency table. The R programs that implement the proposed algorithms are presented in Appendix B. [PUBLICATION ABSTRACT]
Author Fu, Lianyan
Gao, Wei
Shi, Ning-Zhong
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  fullname: Fu, Lianyan
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  givenname: Wei
  surname: Gao
  fullname: Gao, Wei
  organization: School of Mathematics and Statistics, Jilin, China Northeast Normal University
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  givenname: Ning-Zhong
  surname: Shi
  fullname: Shi, Ning-Zhong
  organization: School of Mathematics and Statistics, Jilin, China Northeast Normal University
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Cites_doi 10.1214/aoms/1177692379
10.1016/j.csda.2009.10.017
10.1214/aos/1176347279
10.1093/biomet/55.1.179
10.1103/PhysRev.106.620
10.1016/S0304-4076(01)00110-5
10.1111/j.2517-6161.1962.tb00457.x
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10.1007/BF02530498
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10.1103/PhysRev.108.171
10.2307/2531643
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Jaynes, E. T. (1957a), Information theory and statistical mechanics, Physics Review 106, 620-630.
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Ireland, C. T. and S. Kullback (1968), Contingency tables with given marginals, Biometrika 55, 179-188.
McDonald, J. W. and I. Diamond (1990), On the fitting of generalized linear models with nonnegativity parameter constraints, Biometrics 46, 201-206.
Ruschendorf, L. (1995), Convergence of the iterative proportional fitting procedure, The Annals of Statistics 23, 1160-1174.
Darroch, J. N. (1962), Interactions in multifactor contingency tables, Journal of the Royal Statistical Society Series B. 24, 251-263
Deming, W. E. and F. F. Stephan (1940), On a least squares adjustment of a sampled frequency table when the expected marginal totals are known, The Annals of Mathematical Statistics 11, 427-444.
Golan, A. and J. M. Perlo (2002), Comparison of maximum entropy and higher-order entropy estimators, Journal of Econometrics 107, 1-15.
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Csiszar, I. (1975), I-divergence geometry of probability distributions and minimization problems, The Annals of Probability 3, 146-159.
Gao, W., N. Z. Shi, M. L. Tang, L. Fu and G. Tian (2010), Unified generalized iterative scaling and its applications, Computational Statistics and Data Analysis 54, 1066-1078.
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References_xml – reference: Kullback, S. (1968), Probability densities with given marginals, The Annals of Mathematical Statistics 39, 1236-1243.
– reference: McDonald, J. W. and I. Diamond (1990), On the fitting of generalized linear models with nonnegativity parameter constraints, Biometrics 46, 201-206.
– reference: Ireland, C. T. and S. Kullback (1968), Contingency tables with given marginals, Biometrika 55, 179-188.
– reference: Silvapulle, M. J. and P. K. Sen (2005), Constrained statistical inference: inequality, order and shape constraints, J. Wiley.
– reference: Csiszar, I. (1989), A Geometric interpretation of Darroch and Ratcliff's generalized iterative scaling, The Annals of Statistics 17, 1409-1413.
– reference: Kullback, S. (1967), An extension of information-theoretic derivation of certain limit relations for a Markov chain, SIAM J. Control 5, 51-53.
– reference: Kullback, S. (1959), Information theory and statistics, John Wiley, New York.
– reference: Deming, W. E. and F. F. Stephan (1940), On a least squares adjustment of a sampled frequency table when the expected marginal totals are known, The Annals of Mathematical Statistics 11, 427-444.
– reference: Bhattacharya, B. (2006), An iterative procedure for general probability measures to obtain I-projection onto intersection of convex sets, The Annals of Statistics 34, 878-902.
– reference: Robertson, T., F. T. Wright and R. L. Dykstra (1988), Order restricted statistical inference, John Wiley and Sons, New York.
– reference: Csiszar, I. (1975), I-divergence geometry of probability distributions and minimization problems, The Annals of Probability 3, 146-159.
– reference: Jaynes, E. T. (1957b), Information theory and statistical mechanics II, Physics Review 108, 171-190.
– reference: Ruschendorf, L. (1995), Convergence of the iterative proportional fitting procedure, The Annals of Statistics 23, 1160-1174.
– reference: Ruschendorf, L. and W. Thomsen (1993), Note on the Schrodinger equation and I-projections, Statistics & Probability Letters 17, 369-375.
– reference: Agresti, A. (2002), Categorical data analysis, John Wiley, New York.
– reference: Golan, A. and J. M. Perlo (2002), Comparison of maximum entropy and higher-order entropy estimators, Journal of Econometrics 107, 1-15.
– reference: Jaynes, E. T. (1957a), Information theory and statistical mechanics, Physics Review 106, 620-630.
– reference: Agresti, A. and B. A. Coull (2002), The analysis of contingency tables under inequality constraints, Journal of Statistical Planning and Inference 107, 45-73.
– reference: Gao, W. and N. Z. Shi (2003), I-projection onto isotonic cones and its applications to maximum likelihood estimation for log-linear models, The Annals of Institute of Statistical Mathematics 55, 251-263.
– reference: Gao, W., N. Z. Shi, M. L. Tang, L. Fu and G. Tian (2010), Unified generalized iterative scaling and its applications, Computational Statistics and Data Analysis 54, 1066-1078.
– reference: Darroch, J. N. (1962), Interactions in multifactor contingency tables, Journal of the Royal Statistical Society Series B. 24, 251-263
– reference: Darroch, J. N. and D. Ratcliff (1972), Generalized iterative scaling for log-linear models, The Annals of Mathematical Statistics 43, 1470-1480.
– volume: 107
  start-page: 45
  year: 2002
  end-page: 73
  article-title: The analysis of contingency tables under inequality constraints
  publication-title: Journal of Statistical Planning and Inference
– volume: 55
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  year: 2003
  end-page: 263
  article-title: I–projection onto isotonic cones and its applications to maximum likelihood estimation for log‐linear models
  publication-title: The Annals of Institute of Statistical Mathematics
– volume: 24
  start-page: 251
  year: 1962
  end-page: 263
  article-title: Interactions in multifactor contingency tables
  publication-title: Journal of the Royal Statistical Society Series B
– year: 1959
– volume: 17
  start-page: 369
  year: 1993
  end-page: 375
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  publication-title: Statistics & Probability Letters
– year: 2005
– volume: 11
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  year: 1940
  end-page: 444
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  publication-title: The Annals of Mathematical Statistics
– year: 2002
– year: 1988
– volume: 17
  start-page: 1409
  year: 1989
  end-page: 1413
  article-title: A Geometric interpretation of Darroch and Ratcliff's generalized iterative scaling
  publication-title: The Annals of Statistics
– volume: 107
  start-page: 1
  year: 2002
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  publication-title: Journal of Econometrics
– volume: 5
  start-page: 51
  year: 1967
  end-page: 53
  article-title: An extension of information‐theoretic derivation of certain limit relations for a Markov chain
  publication-title: SIAM J. Control
– volume: 39
  start-page: 1236
  year: 1968
  end-page: 1243
  article-title: Probability densities with given marginals
  publication-title: The Annals of Mathematical Statistics
– volume: 34
  start-page: 878
  year: 2006
  end-page: 902
  article-title: An iterative procedure for general probability measures to obtain I‐projection onto intersection of convex sets
  publication-title: The Annals of Statistics
– volume: 55
  start-page: 179
  year: 1968
  end-page: 188
  article-title: Contingency tables with given marginals
  publication-title: Biometrika
– volume: 106
  start-page: 620
  year: 1957a
  end-page: 630
  article-title: Information theory and statistical mechanics
  publication-title: Physics Review
– volume: 54
  start-page: 1066
  year: 2010
  end-page: 1078
  article-title: Unified generalized iterative scaling and its applications
  publication-title: Computational Statistics and Data Analysis
– volume: 23
  start-page: 1160
  year: 1995
  end-page: 1174
  article-title: Convergence of the iterative proportional fitting procedure
  publication-title: The Annals of Statistics
– volume: 108
  start-page: 171
  year: 1957b
  end-page: 190
  article-title: Information theory and statistical mechanics II
  publication-title: Physics Review
– volume: 46
  start-page: 201
  year: 1990
  end-page: 206
  article-title: On the fitting of generalized linear models with nonnegativity parameter constraints
  publication-title: Biometrics
– volume: 3
  start-page: 146
  year: 1975
  end-page: 159
  article-title: I‐divergence geometry of probability distributions and minimization problems
  publication-title: The Annals of Probability
– volume: 43
  start-page: 1470
  year: 1972
  end-page: 1480
  article-title: Generalized iterative scaling for log‐linear models
  publication-title: The Annals of Mathematical Statistics
– ident: e_1_2_8_8_1
  doi: 10.1214/aoms/1177692379
– ident: e_1_2_8_11_1
  doi: 10.1016/j.csda.2009.10.017
– ident: e_1_2_8_6_1
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– ident: e_1_2_8_13_1
  doi: 10.1093/biomet/55.1.179
– ident: e_1_2_8_14_1
  doi: 10.1103/PhysRev.106.620
– ident: e_1_2_8_12_1
  doi: 10.1016/S0304-4076(01)00110-5
– volume-title: Information theory and statistics
  year: 1959
  ident: e_1_2_8_16_1
– volume: 24
  start-page: 251
  year: 1962
  ident: e_1_2_8_7_1
  article-title: Interactions in multifactor contingency tables
  publication-title: Journal of the Royal Statistical Society Series B
  doi: 10.1111/j.2517-6161.1962.tb00457.x
– ident: e_1_2_8_18_1
  doi: 10.1214/aoms/1177698249
– volume-title: Constrained statistical inference: inequality, order and shape constraints
  year: 2005
  ident: e_1_2_8_23_1
– ident: e_1_2_8_2_1
  doi: 10.1002/0471249688
– ident: e_1_2_8_3_1
  doi: 10.1016/S0378-3758(02)00243-4
– ident: e_1_2_8_21_1
  doi: 10.1214/aos/1176324703
– volume-title: Order restricted statistical inference
  year: 1988
  ident: e_1_2_8_20_1
– ident: e_1_2_8_10_1
  doi: 10.1007/BF02530498
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Snippet This article considers the problem of estimating the cell frequencies in a contingency table under inequality constraints. Algorithms are proposed for cell...
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SubjectTerms Algorithms
Contingency
Convergence
Estimating
Estimating techniques
Frequencies
I-projection
Inequalities
inequality constraints
maximum entropy
optimization
Statistical analysis
Studies
Tables (data)
the minimum discrimination information
Theorems
Title Estimating cell frequencies under inequality constraints based on the Kullback-Leibler information
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https://onlinelibrary.wiley.com/doi/abs/10.1111%2Fj.1467-9574.2011.00513.x
http://econpapers.repec.org/article/blastanee/v_3a66_3ay_3a2012_3ai_3a2_3ap_3a203-216.htm
https://www.proquest.com/docview/993486459
https://www.proquest.com/docview/1022913323
https://www.proquest.com/docview/1762123497
Volume 66
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