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Statistical Papers

, Volume 38, Issue 2, pp 219–229 | Cite as

Contingency tables with prescribed marginals

  • Robert Aebi
Notes

Abstract

For an adjustment of contingency tables to prescribed marginal frequencies Deming and Stephan (1940) minimize a Chi-square expression. Asymptotically equivalently, Ireland and Kullback (1968) minimize a Leibler-Kullback divergence, where the probabilistical arguments for both methods remain vague.

Here we deduce a probabilistical model based on observed contingency tables. It shows that the two above methods and the maximum likelihood approach in Smith (1947) yield asymptotically the ‘most probable’ adjustment under prescribed marginal frequencies.

The fundamental hypothesis of statistical mechanics relates observations to ‘most probable’ realizations. ‘Most probable’ is going to be used in the sense of so-called large deviations. The proposed adjustment has a significant product form and will be generalized to contingency tables with infinitely many cells.

Key Words

contingency tables marginal frequencies large deviations relative entropy non-linear integral equations 

AMS 1990 subject classification

primary 62H17 secondary 62G20, 62G05 

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References

  1. Aebi, R., Nagasawa, M. (1992) Large Deviations and the Propagation of Chaos for Schrödinger Processes. Probab. Theory Relat. Fields94: 53–68CrossRefMathSciNetMATHGoogle Scholar
  2. Aebi, R. (1995) A Solution to Schrödinger's Problem of Non-Linear Integral Equations. Z. angew. Math. Phys. (ZAMP)46: 772–792CrossRefMathSciNetMATHGoogle Scholar
  3. Beurling, A. (1960) An Automorphism of Product Measures. Ann. Math.72: 189–200CrossRefMathSciNetGoogle Scholar
  4. Bickel, P.J., Ritov, Y, Wellner, J.A. (1991) Efficient Estimation of Linear Functionals of a Probability MeasureP with, Known Marginal Distributions. Ann. of Stat.19: 1316–1346CrossRefMathSciNetMATHGoogle Scholar
  5. Boltzmann, L. (1896) Vorlesungen über Gastheorie. J.A. Barth Verlag, LeipzigMATHGoogle Scholar
  6. Carnal, H. (1993) Mathématiques et politique. El. Math.48: 27–32MATHGoogle Scholar
  7. Csiszar, I. (1975)I-Divergence Geometry of Probability Distributions and Minimization Problems. Ann. Probab.3: 146–158CrossRefMathSciNetMATHGoogle Scholar
  8. Csiszar, I. (1984) Sanov Property, GeneralizedI-Projection and a Conditional Limit Theorem. Ann. Probab.12: 768–793CrossRefMathSciNetMATHGoogle Scholar
  9. Deming, W.E., Stephan, F.F. (1940) On a Least Squares Adjustment of a Sampled Frequency Table when the Expected Marginal Totals are Known. Ann. of Math. Stat.11: 427–444CrossRefMathSciNetMATHGoogle Scholar
  10. Fienberg, S.E. (1970) An Iterative Procedure for Estimation in Contingency Tables. Ann. Math. Stat.41: 907–917CrossRefMathSciNetMATHGoogle Scholar
  11. Föllmer, H. (1988) Random Fields and Diffusion Processes. École d'Été de Saint Flour XV–XVII (1985–87). Lecture Notes Math. 1362, Springer-Verlag, BerlinCrossRefGoogle Scholar
  12. Fortet, R. (1940) Résolution d'un Système d'Équation de M. Schrödinger. J. Math. Pures et Appl. IX: 83–95MathSciNetGoogle Scholar
  13. Haberman, S.J. (1984) Adjustment by Minimum Discriminant Information. Ann. of Stat.12: 971–988CrossRefMathSciNetMATHGoogle Scholar
  14. Ireland, C.T., Kullback, S. (1968) Contigency Tables with Given Marginals. Biometrika55: 179–188CrossRefMathSciNetMATHGoogle Scholar
  15. Lanford, O.E. (1973) Entropy and Equilibrium States in Classical Statistical Mechanics. ‘Statistical Mechanics and Mathematical Problems’ (ed. Lenard A.), Lecture Notes in Phys.20, 1–113, Springer-Verlag, BerlinCrossRefGoogle Scholar
  16. Nagasawa, M. (1989) Transformations of Diffusion and Schrödinger Processes. Probab. Th. Rel. Fields82: 109–136CrossRefMathSciNetMATHGoogle Scholar
  17. Nagasawa, M. (1993) Schrödinger Equations and Diffusion Theory. Monographs in Mathematics vol. 86, Birkhäuser-Verlag, BaselMATHGoogle Scholar
  18. Schrödinger, E. (1931) Über die Umkehrung der Naturgesetze. Sitzungsberichte der Preussischen Akademie der Wissenschaften, physikalisch-mathematische Klasse: 144–153Google Scholar
  19. Smith, J.H. (1947) Estimations of Linear Functions of Cell Proportions. Ann. Math. Statist.18: 231–254CrossRefMATHGoogle Scholar

Copyright information

© Springer-Verlag 1997

Authors and Affiliations

  • Robert Aebi
    • 1
  1. 1.Institute of Mathematical StatisticsUniversity of BerneBerneSwitzerland

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