Statistical Science

Comment: Minimalist $g$-Modeling

Roger Koenker and Jiaying Gu

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text


Efron’s elegant approach to $g$-modeling for empirical Bayes problems is contrasted with an implementation of the Kiefer–Wolfowitz nonparametric maximum likelihood estimator for mixture models for several examples. The latter approach has the advantage that it is free of tuning parameters and consequently provides a relatively simple complementary method.

Article information

Statist. Sci., Volume 34, Number 2 (2019), 209-213.

First available in Project Euclid: 19 July 2019

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Nonparametric maximum likelihood mixture model convex optimization


Koenker, Roger; Gu, Jiaying. Comment: Minimalist $g$-Modeling. Statist. Sci. 34 (2019), no. 2, 209--213. doi:10.1214/19-STS706.

Export citation


  • Andersen, E. D. (2010). The Mosek Optimization Tools Manual, Version 6.0. Available from:
  • Deely, J. J. and Lindley, D. V. (1981). Bayes empirical Bayes. J. Amer. Statist. Assoc. 76 833–841.
  • Efron, B. (2010). Large-Scale Inference: Empirical Bayes Methods for Estimation, Testing, and Prediction. Institute of Mathematical Statistics (IMS) Monographs 1. Cambridge Univ. Press, Cambridge.
  • Efron, B. (2016). Empirical Bayes deconvolution estimates. Biometrika 103 1–20.
  • Friberg, H. A. (2012). Users Guide to the R-to-Mosek Interface. Available at
  • Heckman, J. and Singer, B. (1984). A method for minimizing the impact of distributional assumptions in econometric models for duration data. Econometrica 52 271–320.
  • Jiang, W. and Zhang, C.-H. (2009). General maximum likelihood empirical Bayes estimation of normal means. Ann. Statist. 37 1647–1684.
  • Kiefer, J. and Wolfowitz, J. (1956). Consistency of the maximum likelihood estimator in the presence of infinitely many incidental parameters. Ann. Math. Stat. 27 887–906.
  • Koenker, R. and Gu, J. (2015). REBayes: An R Package for Empirical Bayes Methods. Available from
  • Koenker, R. and Mizera, I. (2014). Convex optimization, shape constraints, compound decisions, and empirical Bayes rules. J. Amer. Statist. Assoc. 109 674–685.
  • Laird, N. (1978). Nonparametric maximum likelihood estimation of a mixed distribution. J. Amer. Statist. Assoc. 73 805–811.
  • Lindley, D. V. and Smith, A. (1995). A conversation with Dennis Lindley. Statist. Sci. 10 305–319.
  • Neal, R. M. (2000). Markov chain sampling methods for Dirichlet process mixture models. J. Comput. Graph. Statist. 9 249–265.
  • Robbins, H. (1950). A generalization of the method of maximum likelihood; estimating a mixing distribution (abstract). Ann. Math. Stat. 21 314–315.
  • Ross, G. J. and Markwick, D. (2018). Dirichletprocess: An R Package for Fitting Complex Bayesian Nonparametric Models. Available at
  • Stefanski, L. and Carroll, R. J. (1990). Deconvoluting kernel density estimators. Statistics 21 169–184.

See also