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EM-type algorithms for computing restricted MLEs in multivariate normal distributions and multivariate t-distributions


  • Tian, Guo-Liang
  • Ng, Kai Wang
  • Tan, Ming


Constrained parameter problems arise in a variety of statistical applications but they have been most resistant to solution. This paper proposes methodology for estimating restricted parameters in multivariate normal distributions with known or unknown covariance matrix. The proposed method thus provides a solution to an open problem to find penalized estimation for linear inverse problem with positivity restrictions [Vardi, Y., Lee, D. 1993. From image deblurring to optimal investments: Maximum likelihood solutions for positive linear inverse problems (with discussion). Journal of the Royal Statistical Society, Series B 55, 569-612]. By first considering the simplest bound constraints and then generalizing them to linear inequality constraints, we propose a unified EM-type algorithm for estimating constrained parameters via data augmentation. The key idea is to introduce a sequence of latent variables such that the complete-data model belongs to the exponential family, hence, resulting in a simple E-step and an explicit M-step. Furthermore, we extend restricted multivariate normal distribution to multivariate t-distribution with constrained parameters to obtain robust estimation. With the proposed algorithms, standard errors can be calculated by bootstrapping. The proposed method is appealing for its simplicity and ease of implementation and its applicability to a wide class of parameter restrictions. Three real data sets are analyzed to illustrate different aspects of the proposed methods. Finally, the proposed algorithm is applied to linear inverse problems with possible negativity restrictions and is evaluated numerically.

Suggested Citation

  • Tian, Guo-Liang & Ng, Kai Wang & Tan, Ming, 2008. "EM-type algorithms for computing restricted MLEs in multivariate normal distributions and multivariate t-distributions," Computational Statistics & Data Analysis, Elsevier, vol. 52(10), pages 4768-4778, June.
  • Handle: RePEc:eee:csdana:v:52:y:2008:i:10:p:4768-4778

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    References listed on IDEAS

    1. Moshe Shaked & Nozer Singpurwalla, 1990. "A Bayesian approach for quantile and response probability estimation with applications to reliability," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 42(1), pages 1-19, March.
    2. Mauro Gasparini & Jeffrey Eisele, 2000. "A Curve-Free Method for Phase I Clinical Trials," Biometrics, The International Biometric Society, vol. 56(2), pages 609-615, June.
    3. Vassilis A. Hajivassiliou & Daniel L. McFadden, 1998. "The Method of Simulated Scores for the Estimation of LDV Models," Econometrica, Econometric Society, vol. 66(4), pages 863-896, July.
    4. Donald Rubin & Dorothy Thayer, 1982. "EM algorithms for ML factor analysis," Psychometrika, Springer;The Psychometric Society, vol. 47(1), pages 69-76, March.
    5. Shi, Ning-Zhong & Zheng, Shu-Rong & Guo, Jianhua, 2005. "The restricted EM algorithm under inequality restrictions on the parameters," Journal of Multivariate Analysis, Elsevier, vol. 92(1), pages 53-76, January.
    6. Liu, Chuanhai, 1997. "ML Estimation of the MultivariatetDistribution and the EM Algorithm," Journal of Multivariate Analysis, Elsevier, vol. 63(2), pages 296-312, November.
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    Cited by:

    1. Klugkist, Irene & Hoijtink, Herbert, 2009. "Obtaining similar null distributions in the normal linear model using computational methods," Computational Statistics & Data Analysis, Elsevier, vol. 53(4), pages 877-888, February.
    2. repec:eee:csdana:v:117:y:2018:i:c:p:194-206 is not listed on IDEAS
    3. repec:spr:compst:v:32:y:2017:i:2:d:10.1007_s00180-016-0657-3 is not listed on IDEAS
    4. Ding, Jieli & Tian, Guo-Liang & Yuen, Kam Chuen, 2015. "A new MM algorithm for constrained estimation in the proportional hazards model," Computational Statistics & Data Analysis, Elsevier, vol. 84(C), pages 135-151.

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