Estimating cell probabilities in contingency tables with constraints on marginals/conditionals by geometric programming with applications

Contingency tables are often used to display the multivariate frequency distribution of variables of interest. Under the common multinomial assumption, the first step of contingency table analysis is to estimate cell probabilities. It is well known that the unconstrained maximum likelihood estimator...

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Bibliographic Details
Published inComputational statistics Vol. 30; no. 1; pp. 107 - 129
Main Authors Wang, Xinlei, Lim, Johan, Kim, Seung-Jean, Hahn, Kyu S.
Format Journal Article
LanguageEnglish
Published Berlin/Heidelberg Springer Berlin Heidelberg 01.03.2015
Springer Nature B.V
Subjects
Contingency
Contingency tables
Counting
Economic Theory/Quantitative Economics/Mathematical Methods
Estimators
Exact solutions
Inequality
Marginal analysis
Mathematical analysis
Mathematical models
Mathematics and Statistics
Methods
Multivariate analysis
Optimization
Original Paper
Probability
Probability and Statistics in Computer Science
Probability Theory and Stochastic Processes
Programming
Random variables
Statistics
Studies
Tables
Posynomial
Ordered marginals
Judgement post-stratification
Known marginals
Monomial
Stochastic ordering
Multinomial distribution
Ordered conditionals
Matlab
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Summary:Contingency tables are often used to display the multivariate frequency distribution of variables of interest. Under the common multinomial assumption, the first step of contingency table analysis is to estimate cell probabilities. It is well known that the unconstrained maximum likelihood estimator (MLE) is given by cell counts divided by the total number of observations. However, in the presence of (complex) constraints on the unknown cell probabilities or their functions, the MLE or other types of estimators may often have no closed form and have to be obtained numerically. In this paper, we focus on finding the MLE of cell probabilities in contingency tables under two common types of constraints: known marginals and ordered marginals/conditionals, and propose a novel approach based on geometric programming. We present two important applications that illustrate the usefulness of our approach via comparison with existing methods. Further, we show that our GP-based approach is flexible, readily implementable, effort-saving and can provide a unified framework for various types of constrained estimation of cell probabilities in contingency tables.
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ISSN:0943-4062
1613-9658
DOI:10.1007/s00180-014-0525-y