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Efficient ML Estimation of the Multivariate Normal Distribution from Incomplete Data

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  • Liu, Chuanhai

Abstract

It is well known that the maximum likelihood estimates (MLEs) of a multivariate normal distribution from incomplete data with a monotone pattern have closed-form expressions and that the MLEs from incomplete data with a general missing-data pattern can be obtained using the Expectation-Maximization (EM) algorithm. This article gives closed-form expressions, analogous to the extension of the Bartlett decomposition, for both the MLEs of the parameters and the associated Fisher information matrix from incomplete data with a monotone missing-data pattern. For MLEs of the parameters from incomplete data with a general missing-data pattern, we implement EM and Expectation-Constrained-Maximization-Either (ECME), by augmenting the observed data into a complete monotone sample. We also provide a numerical example, which shows that the monotone EM (MEM) and monotone ECME (MECME) algorithms converge much faster than the EM algorithm.

Suggested Citation

  • Liu, Chuanhai, 1999. "Efficient ML Estimation of the Multivariate Normal Distribution from Incomplete Data," Journal of Multivariate Analysis, Elsevier, vol. 69(2), pages 206-217, May.
    • Handle: RePEc:eee:jmvana:v:69:y:1999:i:2:p:206-217
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    References listed on IDEAS

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    1. Liu, C. H., 1993. "Bartlett's Decomposition of the Posterior Distribution of the Covariance for Normal Monotone Ignorable Missing Data," Journal of Multivariate Analysis, Elsevier, vol. 46(2), pages 198-206, August.
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    Cited by:

    1. Xiaoqian Sun & Dongchu Sun, 2007. "Estimation of a Multivariate Normal Covariance Matrix with Staircase Pattern Data," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 59(2), pages 211-233, June.
    2. Ding, Peng, 2014. "Bayesian robust inference of sample selection using selection-t models," Journal of Multivariate Analysis, Elsevier, vol. 124(C), pages 451-464.
    3. Wan-Lun Wang & Min Liu & Tsung-I Lin, 2017. "Robust skew-t factor analysis models for handling missing data," Statistical Methods & Applications, Springer;Società Italiana di Statistica, vol. 26(4), pages 649-672, November.
    4. Tang, Man-Lai & Wang Ng, Kai & Tian, Guo-Liang & Tan, Ming, 2007. "On improved EM algorithm and confidence interval construction for incomplete rxc tables," Computational Statistics & Data Analysis, Elsevier, vol. 51(6), pages 2919-2933, March.
    5. Sun, Dongchu & Sun, Xiaoqian, 2006. "Estimation of multivariate normal covariance and precision matrices in a star-shape model with missing data," Journal of Multivariate Analysis, Elsevier, vol. 97(3), pages 698-719, March.
    6. Ng, Kai Wang & Tang, Man-Lai & Tan, Ming & Tian, Guo-Liang, 2008. "Grouped Dirichlet distribution: A new tool for incomplete categorical data analysis," Journal of Multivariate Analysis, Elsevier, vol. 99(3), pages 490-509, March.
    7. Lin, Tsung I. & Ho, Hsiu J. & Chen, Chiang L., 2009. "Analysis of multivariate skew normal models with incomplete data," Journal of Multivariate Analysis, Elsevier, vol. 100(10), pages 2337-2351, November.
    8. Wan-Lun Wang & Tsung-I Lin, 2022. "Robust clustering via mixtures of t factor analyzers with incomplete data," Advances in Data Analysis and Classification, Springer;German Classification Society - Gesellschaft für Klassifikation (GfKl);Japanese Classification Society (JCS);Classification and Data Analysis Group of the Italian Statistical Society (CLADAG);International Federation of Classification Societies (IFCS), vol. 16(3), pages 659-690, September.

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