A linearization procedure and a VDM/ECM algorithm for penalized and constrained nonparametric maximum likelihood estimation for mixture models

Suppose independent observations X i , i = 1 , … , n are observed from a mixture model f ( x ; Q ) ≡ ∫ f ( x ; λ ) d Q ( λ ) , where λ is a scalar and Q ( λ ) is a nondegenerate distribution with an unspecified form. We consider to estimate Q ( λ ) by nonparametric maximum likelihood (NPML) method u...

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Published in	Computational statistics & data analysis Vol. 51; no. 6; pp. 2946 - 2957
Main Author	Wang, Ji-Ping
Format	Journal Article
Language	English
Published	Amsterdam Elsevier B.V 01.03.2007 Elsevier Science Elsevier
Series	Computational Statistics & Data Analysis
Subjects	Computing algorithm Constrained NPMLE Exact sciences and technology General topics Mathematics Mixture models Multivariate analysis Nonparametric maximum likelihood Numerical analysis Numerical analysis. Scientific computation Numerical methods in probability and statistics Parametric inference Penalized NPMLE Probability and statistics Sciences and techniques of general use Statistics VDM/ECM Constrained NPMLE Nonparametric maximum likelihood Computing algorithm VDM/ECM Mixture models Penalized NPMLE Mixed distribution Data analysis Linear function Non parametric method Non parametric estimation Constrained estimation Algorithm Penalty method Statistical computation Distribution function Maximum likelihood Linear functional Likelihood function Linearization
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ISSN	0167-9473 1872-7352
DOI	10.1016/j.csda.2006.11.033

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Abstract	Suppose independent observations X i , i = 1 , … , n are observed from a mixture model f ( x ; Q ) ≡ ∫ f ( x ; λ ) d Q ( λ ) , where λ is a scalar and Q ( λ ) is a nondegenerate distribution with an unspecified form. We consider to estimate Q ( λ ) by nonparametric maximum likelihood (NPML) method under two scenarios: (1) the likelihood is penalized by a functional g ( Q ) ; and (2) Q is under a constraint g ( Q ) = g 0 . We propose a simple and reliable algorithm termed VDM/ECM for Q-estimation when the likelihood is penalized by a linear functional. We show this algorithm can be applied to a more general situation where the penalty is not linear, but a function of linear functionals by a linearization procedure. The constrained NPMLE can be found by penalizing the quadratic distance \| g ( Q ) - g 0 \| 2 under a large penalty factor γ > 0 using this algorithm. The algorithm is illustrated with two real data sets.
AbstractList	Suppose independent observations X i , i = 1 , … , n are observed from a mixture model f ( x ; Q ) ≡ ∫ f ( x ; λ ) d Q ( λ ) , where λ is a scalar and Q ( λ ) is a nondegenerate distribution with an unspecified form. We consider to estimate Q ( λ ) by nonparametric maximum likelihood (NPML) method under two scenarios: (1) the likelihood is penalized by a functional g ( Q ) ; and (2) Q is under a constraint g ( Q ) = g 0 . We propose a simple and reliable algorithm termed VDM/ECM for Q-estimation when the likelihood is penalized by a linear functional. We show this algorithm can be applied to a more general situation where the penalty is not linear, but a function of linear functionals by a linearization procedure. The constrained NPMLE can be found by penalizing the quadratic distance \| g ( Q ) - g 0 \| 2 under a large penalty factor γ > 0 using this algorithm. The algorithm is illustrated with two real data sets.
Author	Wang, Ji-Ping
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Keywords	Constrained NPMLE Nonparametric maximum likelihood Computing algorithm VDM/ECM Mixture models Penalized NPMLE Mixed distribution Data analysis Linear function Non parametric method Non parametric estimation Constrained estimation Algorithm Penalty method Statistical computation Distribution function Maximum likelihood Linear functional Likelihood function Linearization
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SubjectTerms	Computing algorithm Constrained NPMLE Exact sciences and technology General topics Mathematics Mixture models Multivariate analysis Nonparametric maximum likelihood Numerical analysis Numerical analysis. Scientific computation Numerical methods in probability and statistics Parametric inference Penalized NPMLE Probability and statistics Sciences and techniques of general use Statistics VDM/ECM
Title	A linearization procedure and a VDM/ECM algorithm for penalized and constrained nonparametric maximum likelihood estimation for mixture models
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