Recent progress in log-concave density estimation. (English) Zbl 1407.62126

Samworth, Richard J.

Stat. Sci. 33, No. 4, 493-509 (2018).

Summary: 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.

MSC:

62G07	Density estimation
62G05	Nonparametric estimation
62E10	Characterization and structure theory of statistical distributions

Keywords:

log-concavity; maximum likelihood estimation

Software:

SolvOpt; logcondens; scar; logcondiscr; logcondens.mode; logconcens; LogConcDEAD; FastICA

References:

[1]	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.
[2]	Balabdaoui, F. and Doss, C. R. (2018). Inference for a two-component mixture of symmetric distributions under log-concavity. Bernoulli24 1053-1071. · Zbl 1419.62059
[3]	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. · Zbl 1160.62008
[4]	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. · Zbl 1411.62119
[5]	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.
[6]	Baraud, Y. and Birgé, L. (2016). Rho-estimators for shape restricted density estimation. Stochastic Process. Appl.126 3888-3912. · Zbl 1419.62070
[7]	Birgé, L. (1989). The Grenander estimator: A nonasymptotic approach. Ann. Statist.17 1532-1549. · Zbl 0703.62042
[8]	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. · Zbl 1093.52003
[9]	Chang, G. T. and Walther, G. (2007). Clustering with mixtures of log-concave distributions. Comput. Statist. Data Anal.51 6242-6251. · Zbl 1445.62141
[10]	Chen, Y. and Samworth, R. J. (2013). Smoothed log-concave maximum likelihood estimation with applications. Statist. Sinica23 1373-1398. · Zbl 06202711
[11]	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. · Zbl 1414.62153
[12]	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.
[13]	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. · Zbl 1329.62183
[14]	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. · Zbl 1329.62183
[15]	Dharmadhikari, S. and Joag-Dev, K. (1988). Unimodality, Convexity, and Applications. Academic Press, Boston, MA. · Zbl 0646.62008
[16]	Doss, C. R. and Wellner, J. A. (2016a). Inference for the mode of a log-concave density. https://arxiv.org/abs/1611.10348. · Zbl 1439.62098
[17]	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. · Zbl 1338.62101
[18]	Dudley, R. M. (2002). Real Analysis and Probability. Cambridge Studies in Advanced Mathematics74. Cambridge Univ. Press, Cambridge. · Zbl 1023.60001
[19]	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 https://arxiv.org/abs/0707.4643v4.
[20]	Dümbgen, L. and Rufibach, K. (2009). Maximum likelihood estimation of a log-concave density and its distribution function: Basic properties and uniform consistency. Bernoulli15 40-68. · Zbl 1200.62030
[21]	Dümbgen, L. and Rufibach, K. (2011). logcondens: Computations related to univariate log-concave density estimation. J. Stat. Softw.39 1-28.
[22]	Dümbgen, L., Rufibach, K. and Schuhmacher (2013). logconcens: Maximum likelihood estimation of a log-concave density based on censored data. R package available at: https://cran.r-project.org/web/packages/logconcens/index.html. · Zbl 1298.62062
[23]	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. · Zbl 1298.62062
[24]	Dümbgen, L., Samworth, R. and Schuhmacher, D. (2011). Approximation by log-concave distributions, with applications to regression. Ann. Statist.39 702-730. · Zbl 1216.62023
[25]	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. · Zbl 1327.62204
[26]	Eilers, P. H. C. and Borgdorff, M. W. (2007). Non-parametric log-concave mixtures. Comput. Statist. Data Anal.51 5444-5451. · Zbl 1445.62070
[27]	Eriksson, J. and Koivunen, V. (2004). Identifiability, separability and uniqueness of linear ICA models. IEEE Signal Process. Lett.11 601-604.
[28]	Gao, F. and Wellner, J. A. (2017). Entropy of convex functions on \(\mathbb{R}^d \). Constr. Approx.46 565-592. · Zbl 1381.52016
[29]	Grenander, U. (1956). On the theory of mortality measurement. II. Skand. Aktuarietidskr.39 125-153. · Zbl 0077.33715
[30]	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. · Zbl 1373.62144
[31]	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. · Zbl 1043.62026
[32]	Groeneboom, P., Jongbloed, G. and Wellner, J. A. (2001b). Estimation of a convex function: Characterizations and asymptotic theory. Ann. Statist.29 1653-1698. · Zbl 1043.62027
[33]	Han, Q. and Wellner, J. A. (2016a). Approximation and estimation of \(s\)-concave densities via Rényi divergences. Ann. Statist.44 1332-1359. · Zbl 1338.62105
[34]	Han, Q. and Wellner, J. A. (2016b). Multivariate convex regression: Global risk bounds and adaptation. Available at https://arxiv.org/abs/1601.06844.
[35]	Henningsson, T. and Åström, K. J. (2006). Log-concave observers. In Proc. 17th International Symposium on Mathematical Theory of Networks and Systems.
[36]	Hunter, D. R., Wang, S. and Hettmansperger, T. P. (2007). Inference for mixtures of symmetric distributions. Ann. Statist.35 224-251. · Zbl 1114.62035
[37]	Hyvärinen, A., Karhunen, J. and Oja, E. (2001). Independent Component Analysis. Wiley, Hoboken, New Jersey.
[38]	Ibragimov, I. A. (1956). On the composition of unimodal distributions. Theory Probab. Appl.1 255-260.
[39]	Kappel, F. and Kuntsevich, A. V. (2000). An implementation of Shor’s \(r\)-algorithm. Comput. Optim. Appl.15 193-205. · Zbl 0947.90112
[40]	Kim, A. K. H. and Samworth, R. J. (2016). Global rates of convergence in log-concave density estimation. Ann. Statist.44 2756-2779. · Zbl 1360.62157
[41]	Kim, A. K. H., Guntuboyina, A. and Samworth, R. J. (2018). Adaptation in log-concave density estimation. Ann. Statist. To appear. · Zbl 1408.62062
[42]	Koenker, R. and Mizera, I. (2010). Quasi-concave density estimation. Ann. Statist.38 2998-3027. · Zbl 1200.62031
[43]	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.
[44]	Müller, S. and Rufibach, K. (2009). Smooth tail-index estimation. J. Stat. Comput. Simul.79 1155-1167. · Zbl 1179.62075
[45]	Prakasa Rao, B. L. S. (1969). Estimation of a unimodal density. Sankhyā Ser. A31 23-36. · Zbl 0181.45901
[46]	Prékopa, A. (1973). Contributions to the theory of stochastic programming. Math. Program.4 202-221. · Zbl 0273.90045
[47]	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.
[48]	Samworth, R. J. and Yuan, M. (2012). Independent component analysis via nonparametric maximum likelihood estimation. Ann. Statist.40 2973-3002. · Zbl 1296.62084
[49]	Saumard, A. and Wellner, J. A. (2014). Log-concavity and strong log-concavity: A review. Stat. Surv.8 45-114. · Zbl 1360.62055
[50]	Schuhmacher, D., Hüsler, A. and Dümbgen, L. (2011). Multivariate log-concave distributions as a nearly parametric model. Stat. Risk Model.28 277-295. · Zbl 1245.62060
[51]	Seregin, A. and Wellner, J. A. (2010). Nonparametric estimation of multivariate convex-transformed densities. Ann. Statist.38 3751-3781. · Zbl 1204.62058
[52]	Strassen, V. (1965). The existence of probability measures with given marginals. Ann. Math. Stat.36 423-439. · Zbl 0135.18701
[53]	van Eeden, C. (1958). Testing and Estimating Ordered Parameters of Probability Distributions. Mathematical Centre, Amsterdam. · Zbl 0086.35302
[54]	Walther, G. (2002). Detecting the presence of mixing with multiscale maximum likelihood. J. Amer. Statist. Assoc.97 508-513. · Zbl 1073.62533
[55]	Walther, G. (2009). Inference and modeling with log-concave distributions. Statist. Sci.24 319-327. · Zbl 1329.62192
[56]	Xu, M.