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A computational strategy for doubly smoothed MLE exemplified in the normal mixture model

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  • Seo, Byungtae
  • Lindsay, Bruce G.

Abstract

A typical problem for the parameter estimation in normal mixture models is an unbounded likelihood and the presence of many spurious local maxima. To resolve this problem, we apply the doubly smoothed maximum likelihood estimator (DS-MLE) proposed by Seo and Lindsay (in preparation). We discuss the computational issues of the DS-MLE and propose a simulation-based DS-MLE using Monte Carlo methods as a general computational tool. Simulation results show that the DS-MLE is virtually consistent for any bandwidth choice. Moreover, the parameter estimates in the DS-MLE are quite robust to the choice of bandwidths, as the theory indicates. A new method for the bandwidth selection is also proposed.

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  • Seo, Byungtae & Lindsay, Bruce G., 2010. "A computational strategy for doubly smoothed MLE exemplified in the normal mixture model," Computational Statistics & Data Analysis, Elsevier, vol. 54(8), pages 1930-1941, August.
    • Handle: RePEc:eee:csdana:v:54:y:2010:i:8:p:1930-1941
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    1. Gabriela Ciuperca & Andrea Ridolfi & Jérôme Idier, 2003. "Penalized Maximum Likelihood Estimator for Normal Mixtures," Scandinavian Journal of Statistics, Danish Society for Theoretical Statistics;Finnish Statistical Society;Norwegian Statistical Association;Swedish Statistical Association, vol. 30(1), pages 45-59, March.
    2. D. Böhning, 1986. "A vertex-exchange-method in D-optimal design theory," Metrika: International Journal for Theoretical and Applied Statistics, Springer, vol. 33(1), pages 337-347, December.
    3. repec:dau:papers:123456789/6069 is not listed on IDEAS
    4. Chen, Jiahua & Tan, Xianming, 2009. "Inference for multivariate normal mixtures," Journal of Multivariate Analysis, Elsevier, vol. 100(7), pages 1367-1383, August.
    5. Hall, Peter, 1990. "Using the bootstrap to estimate mean squared error and select smoothing parameter in nonparametric problems," Journal of Multivariate Analysis, Elsevier, vol. 32(2), pages 177-203, February.
    6. Ingrassia, Salvatore & Rocci, Roberto, 2007. "Constrained monotone EM algorithms for finite mixture of multivariate Gaussians," Computational Statistics & Data Analysis, Elsevier, vol. 51(11), pages 5339-5351, July.
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    Cited by:

    1. Kim, Daeyoung & Seo, Byungtae, 2014. "Assessment of the number of components in Gaussian mixture models in the presence of multiple local maximizers," Journal of Multivariate Analysis, Elsevier, vol. 125(C), pages 100-120.
    2. Seo, Byungtae & Kim, Daeyoung, 2012. "Root selection in normal mixture models," Computational Statistics & Data Analysis, Elsevier, vol. 56(8), pages 2454-2470.
    3. Seo, Byungtae & Lindsay, Bruce G., 2013. "Nearly universal consistency of maximum likelihood in discrete models," Statistics & Probability Letters, Elsevier, vol. 83(7), pages 1699-1702.
    4. Hampel, Frank & Hennig, Christian & Ronchetti, Elvezio, 2011. "A smoothing principle for the Huber and other location M-estimators," Computational Statistics & Data Analysis, Elsevier, vol. 55(1), pages 324-337, January.
    5. Seo, Byungtae, 2017. "The doubly smoothed maximum likelihood estimation for location-shifted semiparametric mixtures," Computational Statistics & Data Analysis, Elsevier, vol. 108(C), pages 27-39.
    6. Ingrassia, Salvatore & Rocci, Roberto, 2011. "Degeneracy of the EM algorithm for the MLE of multivariate Gaussian mixtures and dynamic constraints," Computational Statistics & Data Analysis, Elsevier, vol. 55(4), pages 1715-1725, April.
    7. Roberto Mari & Roberto Rocci & Stefano Antonio Gattone, 2020. "Scale-constrained approaches for maximum likelihood estimation and model selection of clusterwise linear regression models," Statistical Methods & Applications, Springer;Società Italiana di Statistica, vol. 29(1), pages 49-78, March.
    8. Chee, Chew-Seng & Wang, Yong, 2013. "Minimum quadratic distance density estimation using nonparametric mixtures," Computational Statistics & Data Analysis, Elsevier, vol. 57(1), pages 1-16.

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