Convex Optimization, Shape Constraints, Compound Decisions, and Empirical Bayes Rules
Estimation of mixture densities for the classical Gaussian compound decision problem and their associated (empirical) Bayes rules is considered from two new perspectives. The first, motivated by Brown and Greenshtein, introduces a nonparametric maximum likelihood estimator of the mixture density sub...
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| Published in: | Journal of the American Statistical Association Vol. 109; no. 506; pp. 674 - 685 |
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| Main Authors: | , |
| Format: | Journal Article |
| Language: | English |
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Alexandria
Taylor & Francis
01.06.2014
Taylor & Francis Group, LLC Taylor & Francis Ltd |
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| ISSN: | 1537-274X, 0162-1459, 1537-274X |
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| Abstract | Estimation of mixture densities for the classical Gaussian compound decision problem and their associated (empirical) Bayes rules is considered from two new perspectives. The first, motivated by Brown and Greenshtein, introduces a nonparametric maximum likelihood estimator of the mixture density subject to a monotonicity constraint on the resulting Bayes rule. The second, motivated by Jiang and Zhang, proposes a new approach to computing the Kiefer–Wolfowitz nonparametric maximum likelihood estimator for mixtures. In contrast to prior methods for these problems, our new approaches are cast as convex optimization problems that can be efficiently solved by modern interior point methods. In particular, we show that the reformulation of the Kiefer–Wolfowitz estimator as a convex optimization problem reduces the computational effort by several orders of magnitude for typical problems , by comparison to prior EM-algorithm based methods, and thus greatly expands the practical applicability of the resulting methods. Our new procedures are compared with several existing empirical Bayes methods in simulations employing the well-established design of Johnstone and Silverman. Some further comparisons are made based on prediction of baseball batting averages. A Bernoulli mixture application is briefly considered in the penultimate section. |
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| AbstractList | Estimation of mixture densities for the classical Gaussian compound decision problem and their associated (empirical) Bayes rules is considered from two new perspectives. The first, motivated by Brown and Greenshtein, introduces a nonparametric maximum likelihood estimator of the mixture density subject to a monotonicity constraint on the resulting Bayes rule. The second, motivated by Jiang and Zhang, proposes a new approach to computing the Kiefer-Wolfowitz nonparametric maximum likelihood estimator for mixtures. In contrast to prior methods for these problems, our new approaches are cast as convex optimization problems that can be efficiently solved by modern interior point methods. In particular, we show that the reformulation of the Kiefer-Wolfowitz estimator as a convex optimization problem reduces the computational effort by several orders of magnitude for typical problems, by comparison to prior EM-algorithm based methods, and thus greatly expands the practical applicability of the resulting methods. Our new procedures are compared with several existing empirical Bayes methods in simulations employing the well-established design of Johnstone and Silverman. Some further comparisons are made based on prediction of baseball batting averages. A Bernoulli mixture application is briefly considered in the penultimate section. |
| Author | Koenker, Roger Mizera, Ivan |
| Author_xml | – sequence: 1 givenname: Roger surname: Koenker fullname: Koenker, Roger – sequence: 2 givenname: Ivan surname: Mizera fullname: Mizera, Ivan |
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| Cites_doi | 10.1111/j.1467-9574.1983.tb00796.x 10.1006/aima.1994.1002 10.2140/pjm.1971.39.439 10.1214/aoms/1177728066 10.1214/10-IMSCOLL618 10.1016/0304-4076(84)90075-7 10.1214/09-AOS790 10.1198/jasa.2011.tm11181 10.1111/j.1467-9469.2007.00585.x 10.1214/08-AOS630 10.1515/9781400873173 10.1111/j.1467-9469.2007.00588.x 10.1214/10-AOS814 10.1214/10-AOS840 10.1214/009053604000000030 10.1017/CBO9780511761362 10.1214/aos/1015345958 10.1016/j.csda.2006.12.013 10.1214/08-AOS638 10.1111/j.1467-9868.2010.00753.x 10.1080/01621459.1978.10480103 10.1214/07-AOAS138 10.3150/08-BEJ141 10.1214/aos/1176346059 10.1214/aos/1032526949 |
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| References | cit0011 cit0012 cit0031 cit0010 Rockafellar R.T. (cit0025) 1970 Johnstone I. (cit0016) 2004 Koenker R. (cit0018) 2013 Jiang W. (cit0015) 2010 Robbins H. (cit0024) 1956 cit0019 cit0017 cit0013 cit0014 cit0022 cit0023 cit0021 cit0008 cit0009 cit0006 cit0028 cit0007 cit0029 cit0004 cit0026 cit0005 cit0002 cit0003 |
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| SubjectTerms | Algorithms Baseball Bayes estimators Bayes rule Bayesian analysis Bernoulli Hypothesis Constraints Convex analysis Density estimation Empirical Bayes Estimating techniques Estimation methods Estimators Interior points Iterative solutions Kiefer-Wolfowitz maximum likelihood estimator Maximum likelihood estimation Mixture models Mixtures Normal distribution Objective functions Optimization prediction Rules Scientific method Statistics system optimization Theory and Methods |
| Title | Convex Optimization, Shape Constraints, Compound Decisions, and Empirical Bayes Rules |
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