×

zbMATH — the first resource for mathematics

A note on the IPF algorithm when the marginal problem is unsolvable. (English) Zbl 1245.62070
Summary: We analyze the asymptotic behavior of the IPF algorithm for the problem of finding a \(2\times 2\times 2\) contingency table whose pair marginals are all equal to a specified \(2\times 2\) table, depending on a parameter. When this parameter lies below a certain threshold the marginal problem has no solution. We show that in this case the IPF has a “period three limit cycle” attracting all positive initial tables, and a bifurcation occurs when the parameter crosses the threshold.

MSC:
62H17 Contingency tables
65C60 Computational problems in statistics (MSC2010)
PDF BibTeX XML Cite
Full Text: Link EuDML
References:
[1] Csiszár I.: I-divergence geometry of probability distributions and minimization problems. Ann. Probab. 3 (1975), 146-158 · Zbl 0318.60013
[2] Deming W. E., Stephan F. F.: On a least square adjustment of a sampled frequency table when the expected marginal totals are known. Ann. Math. Statist. 11 (1940), 427-444 · Zbl 0024.05502
[3] Haberman S. J.: The analysis of frequency data. The University of Chicago Press, Chicago 1974 · Zbl 0325.62017
[4] Jensen S. T., Johansen, S., Lauritzen S. L.: Globally convergent algorithms for maximizing a likelihood function. Biometrika 78 (1991), 867-877 · Zbl 0752.62031
[5] Jiroušek R.: Solution of the marginal problem and decomposable distributions. Kybernetika 27 (1991), 403-412 · Zbl 0752.60009
[6] Lauritzen S. L.: Graphical Models. Clarendon Press, Oxford 1996 · Zbl 1055.62126
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.