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Degeneracy in the Maximum Likelihood Estimation of Univariate Gaussian Mixtures for Grouped Data and Behaviour of the EM Algorithm

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  • CHRISTOPHE BIERNACKI

Abstract

. In the context of the univariate Gaussian mixture with grouped data, it is shown that the global maximum of the likelihood may correspond to a situation where a Dirac lies in any non‐empty interval. Existence of a domain of attraction near such a maximizer is discussed and we establish that the expectation‐maximization (EM) iterates move extremely slowly inside this domain. These theoretical results are illustrated both by some Monte‐Carlo experiments and by a real data set. To help practitioners identify and discard these potentially dangerous degenerate maximizers, a specific stopping rule for EM is proposed.

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  • Christophe Biernacki, 2007. "Degeneracy in the Maximum Likelihood Estimation of Univariate Gaussian Mixtures for Grouped Data and Behaviour of the EM Algorithm," Scandinavian Journal of Statistics, Danish Society for Theoretical Statistics;Finnish Statistical Society;Norwegian Statistical Association;Swedish Statistical Association, vol. 34(3), pages 569-586, September.
    • Handle: RePEc:bla:scjsta:v:34:y:2007:i:3:p:569-586
    DOI: 10.1111/j.1467-9469.2006.00553.x
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    Cited by:

    1. C. Biernacki & J. Jacques & C. Keribin, 2023. "A Survey on Model-Based Co-Clustering: High Dimension and Estimation Challenges," Journal of Classification, Springer;The Classification Society, vol. 40(2), pages 332-381, July.
    2. Maddalena Cavicchioli, 2016. "Statistical Analysis Of Mixture Vector Autoregressive Models," Scandinavian Journal of Statistics, Danish Society for Theoretical Statistics;Finnish Statistical Society;Norwegian Statistical Association;Swedish Statistical Association, vol. 43(4), pages 1192-1213, December.
    3. Yasutomo Murasawa, 2020. "Measuring public inflation perceptions and expectations in the UK," Empirical Economics, Springer, vol. 59(1), pages 315-344, July.
    4. Murasawa, Yasutomo, 2017. "Measuring the Distributions of Public Inflation Perceptions and Expectations in the UK," MPRA Paper 76244, University Library of Munich, Germany.

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