Statistical Science

Recent Progress in Log-Concave Density Estimation

Richard J. Samworth

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


In recent years, log-concave density estimation via maximum likelihood estimation has emerged as a fascinating alternative to traditional nonparametric smoothing techniques, such as kernel density estimation, which require the choice of one or more bandwidths. The purpose of this article is to describe some of the properties of the class of log-concave densities on $\mathbb{R}^{d}$ which make it so attractive from a statistical perspective, and to outline the latest methodological, theoretical and computational advances in the area.

Article information

Statist. Sci., Volume 33, Number 4 (2018), 493-509.

First available in Project Euclid: 29 November 2018

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Log-concavity maximum likelihood estimation


Samworth, Richard J. Recent Progress in Log-Concave Density Estimation. Statist. Sci. 33 (2018), no. 4, 493--509. doi:10.1214/18-STS666.

Export citation


  • Aleksandrov, A. D. (1939). Almost everywhere existence of the second differential of a convex functions and related properties of convex surfaces. Uchenye Zapisky Leningrad. Gos. Univ. Math. Ser. 37 3–35.
  • Balabdaoui, F. and Doss, C. R. (2018). Inference for a two-component mixture of symmetric distributions under log-concavity. Bernoulli 24 1053–1071.
  • Balabdaoui, F., Rufibach, K. and Wellner, J. A. (2009). Limit distribution theory for maximum likelihood estimation of a log-concave density. Ann. Statist. 37 1299–1331.
  • Balabdaoui, F., Jankowski, H., Rufibach, K. and Pavlides, M. (2013). Asymptotics of the discrete log-concave maximum likelihood estimator and related applications. J. R. Stat. Soc. Ser. B. Stat. Methodol. 75 769–790.
  • Balázs, G., Gyögy, A. and Szepesvári, C. (2015). Near-optimal max-affine estimators for convex regression. In Proc. 18th International Conference on Artificial Intelligence and Statistics (AISTATS) 56–64.
  • Baraud, Y. and Birgé, L. (2016). Rho-estimators for shape restricted density estimation. Stochastic Process. Appl. 126 3888–3912.
  • Birgé, L. (1989). The Grenander estimator: A nonasymptotic approach. Ann. Statist. 17 1532–1549.
  • Brass, P. (2005). On the size of higher-dimensional triangulations. In Combinatorial and Computational Geometry. Math. Sci. Res. Inst. Publ. 52 147–153. Cambridge Univ. Press, Cambridge.
  • Chang, G. T. and Walther, G. (2007). Clustering with mixtures of log-concave distributions. Comput. Statist. Data Anal. 51 6242–6251.
  • Chen, Y. and Samworth, R. J. (2013). Smoothed log-concave maximum likelihood estimation with applications. Statist. Sinica 23 1373–1398.
  • Chen, Y. and Samworth, R. J. (2016). Generalized additive and index models with shape constraints. J. R. Stat. Soc. Ser. B. Stat. Methodol. 78 729–754.
  • Cule, M., Gramacy, R. B. and Samworth, R. (2009). LogConcDEAD: An R package for maximum likelihood estimation of a multivariate log-concave density. J. Stat. Softw. 29.
  • Cule, M. and Samworth, R. (2010). Theoretical properties of the log-concave maximum likelihood estimator of a multidimensional density. Electron. J. Stat. 4 254–270.
  • Cule, M., Samworth, R. and Stewart, M. (2010). Maximum likelihood estimation of a multi-dimensional log-concave density. J. R. Stat. Soc. Ser. B. Stat. Methodol. 72 545–607.
  • Dharmadhikari, S. and Joag-Dev, K. (1988). Unimodality, Convexity, and Applications. Academic Press, Boston, MA.
  • Doss, C. R. and Wellner, J. A. (2016a). Inference for the mode of a log-concave density.
  • Doss, C. R. and Wellner, J. A. (2016b). Global rates of convergence of the MLEs of log-concave and $s$-concave densities. Ann. Statist. 44 954–981.
  • Dudley, R. M. (2002). Real Analysis and Probability. Cambridge Studies in Advanced Mathematics 74. Cambridge Univ. Press, Cambridge.
  • Dümbgen, L., Hüsler, A. and Rufibach, K. (2007). Active set and EM algorithms for log-concave densities based on complete and censored data. Available at
  • Dümbgen, L. and Rufibach, K. (2009). Maximum likelihood estimation of a log-concave density and its distribution function: Basic properties and uniform consistency. Bernoulli 15 40–68.
  • Dümbgen, L. and Rufibach, K. (2011). logcondens: Computations related to univariate log-concave density estimation. J. Stat. Softw. 39 1–28.
  • Dümbgen, L., Rufibach, K. and Schuhmacher (2013). logconcens: Maximum likelihood estimation of a log-concave density based on censored data. R package available at:
  • Dümbgen, L., Rufibach, K. and Schuhmacher, D. (2014). Maximum-likelihood estimation of a log-concave density based on censored data. Electron. J. Stat. 8 1405–1437.
  • Dümbgen, L., Samworth, R. and Schuhmacher, D. (2011). Approximation by log-concave distributions, with applications to regression. Ann. Statist. 39 702–730.
  • Dümbgen, L., Samworth, R. J. and Schuhmacher, D. (2013). Stochastic search for semiparametric linear regression models. In From Probability to Statistics and Back: High-Dimensional Models and Processes. Inst. Math. Stat. (IMS) Collect. 9 78–90. IMS, Beachwood, OH.
  • Eilers, P. H. C. and Borgdorff, M. W. (2007). Non-parametric log-concave mixtures. Comput. Statist. Data Anal. 51 5444–5451.
  • Eriksson, J. and Koivunen, V. (2004). Identifiability, separability and uniqueness of linear ICA models. IEEE Signal Process. Lett. 11 601–604.
  • Gao, F. and Wellner, J. A. (2017). Entropy of convex functions on $\mathbb{R}^{d}$. Constr. Approx. 46 565–592.
  • Grenander, U. (1956). On the theory of mortality measurement. II. Skand. Aktuarietidskr. 39 125–153.
  • Groeneboom, P. (1985). Estimating a monotone density. In Proceedings of the Berkeley Conference in Honor of Jerzy Neyman and Jack Kiefer, Vol. II (Berkeley, Calif., 1983). 539–555. Wadsworth, Belmont, CA.
  • Groeneboom, P., Jongbloed, G. and Wellner, J. A. (2001a). A canonical process for estimation of convex functions: The “invelope” of integrated Brownian motion $+t^{4}$. Ann. Statist. 29 1620–1652.
  • Groeneboom, P., Jongbloed, G. and Wellner, J. A. (2001b). Estimation of a convex function: Characterizations and asymptotic theory. Ann. Statist. 29 1653–1698.
  • Han, Q. and Wellner, J. A. (2016a). Approximation and estimation of $s$-concave densities via Rényi divergences. Ann. Statist. 44 1332–1359.
  • Han, Q. and Wellner, J. A. (2016b). Multivariate convex regression: Global risk bounds and adaptation. Available at
  • Henningsson, T. and Åström, K. J. (2006). Log-concave observers. In Proc. 17th International Symposium on Mathematical Theory of Networks and Systems.
  • Hunter, D. R., Wang, S. and Hettmansperger, T. P. (2007). Inference for mixtures of symmetric distributions. Ann. Statist. 35 224–251.
  • Hyvärinen, A., Karhunen, J. and Oja, E. (2001). Independent Component Analysis. Wiley, Hoboken, New Jersey.
  • Ibragimov, I. A. (1956). On the composition of unimodal distributions. Theory Probab. Appl. 1 255–260.
  • Kappel, F. and Kuntsevich, A. V. (2000). An implementation of Shor’s $r$-algorithm. Comput. Optim. Appl. 15 193–205.
  • Kim, A. K. H. and Samworth, R. J. (2016). Global rates of convergence in log-concave density estimation. Ann. Statist. 44 2756–2779.
  • Kim, A. K. H., Guntuboyina, A. and Samworth, R. J. (2018). Adaptation in log-concave density estimation. Ann. Statist. To appear.
  • Koenker, R. and Mizera, I. (2010). Quasi-concave density estimation. Ann. Statist. 38 2998–3027.
  • Marshall, A. W. (1970). Discussion of Barlow and van Zwet’s paper. In Nonparametric Techniques in Statistical Inference. Proceedings of the First International Symposium on Nonparametric Techniques Held at Indiana University, June 1969. Cambridge Univ. Press, London.
  • Müller, S. and Rufibach, K. (2009). Smooth tail-index estimation. J. Stat. Comput. Simul. 79 1155–1167.
  • Prakasa Rao, B. L. S. (1969). Estimation of a unimodal density. Sankhyā Ser. A 31 23–36.
  • Prékopa, A. (1973). Contributions to the theory of stochastic programming. Math. Program. 4 202–221.
  • Prékopa, A. (1980). Logarithmic concave measures and related topics. In Stochastic Programming (Proc. Internat. Conf., Univ. Oxford, Oxford, 1974) (M. A. H. Dempster ed.) 63–82. Academic Press, London.
  • Samworth, R. J. and Yuan, M. (2012). Independent component analysis via nonparametric maximum likelihood estimation. Ann. Statist. 40 2973–3002.
  • Saumard, A. and Wellner, J. A. (2014). Log-concavity and strong log-concavity: A review. Stat. Surv. 8 45–114.
  • Schuhmacher, D., Hüsler, A. and Dümbgen, L. (2011). Multivariate log-concave distributions as a nearly parametric model. Stat. Risk Model. 28 277–295.
  • Seregin, A. and Wellner, J. A. (2010). Nonparametric estimation of multivariate convex-transformed densities. Ann. Statist. 38 3751–3781.
  • Strassen, V. (1965). The existence of probability measures with given marginals. Ann. Math. Stat. 36 423–439.
  • van Eeden, C. (1958). Testing and Estimating Ordered Parameters of Probability Distributions. Mathematical Centre, Amsterdam.
  • Walther, G. (2002). Detecting the presence of mixing with multiscale maximum likelihood. J. Amer. Statist. Assoc. 97 508–513.
  • Walther, G. (2009). Inference and modeling with log-concave distributions. Statist. Sci. 24 319–327.
  • Xu, M. and Samworth, R. J. (2017). High-dimensional nonparametric density estimation via symmetry and shape constraints. Working paper. Available at: