A Computational Approach to Log-Concave Density Estimation

Open access


Non-parametric density estimation with shape restrictions has witnessed a great deal of attention recently. We consider the maximum-likelihood problem of estimating a log-concave density from a given finite set of empirical data and present a computational approach to the resulting optimization problem. Our approach targets the ability to trade-off computational costs against estimation accuracy in order to alleviate the curse of dimensionality of density estimation in higher dimensions.

[1] Y. Chen and R. J. Samworth, \Smoothed log-concave maximum likelihood estimation with applications," Statist. Sinica, vol. 23, 2013.

[2] M. Cule, R. Samworth, and M. Stewart, \Maximum likelihood estimation of a multi-dimensional logconcave density," J. R. Stat. Soc. Series B Stat. Methodol., vol. 72, no. 5, pp. 545-607, 2010.

[3] L. Dümbgen and K. Rufibach, \Maximum likelihood estimation of a logconcave density and its distribution function: Basic properties and uniform consistency," Bernoulli, vol. 15, no. 1, pp. 40-68, 2009.

[4] R. Koenker and I. Mizera, \Quasi-concave density estimation," Ann. Stat., vol. 38, no. 5, pp. 2998-3027, 2010.

[5] G. Walther, \Inference and modeling with log-concave distributions," Statist. Sci., vol. 24, no. 3, pp. 319-327, 2009.

[6] B. Klartag and V. Milman, \Geometry of log-concave functions and measures," Geom. Dedicata, vol. 112, no. 1, pp. 169-182, 2005.

[7] L. Lovász and S. Vempala, \The geometry of log-concave functions and sampling algorithms," Rand. Structures Alg., vol. 30, no. 3, pp. 307-358, 2007.

[8] A. Seregin and J. Wellner, \Nonparametric estimation of multivariate convex-transformed densities," Ann. Statistics, vol. 38, no. 6, pp. 3751-3781, 2010.

[9] B. Silverman, \On the estimation of a probability density function by the maximum penalized likelihood method," Ann. Stat., vol. 10, no. 3, pp. 795-810, 1982.

[10] R. Rockafellar and R.-B. Wets, Variational Analysis, vol. 317. Springer, 3rd ed., 2009.

[11] M. Cule, R. Gramacy, and R. Samworth, \LogConcDEAD: An R package for maximum likelihood estimation of a multivariate log-concave density," J. Stat. Softw., vol. 29, no. 2, pp. 1-20, 2009.

[12] S. Boyd and L. Vandenberghe, Convex Optimization. Cambridge University Press, 2004.

[13] F. Rathke, S. Schmidt, and C. Schnörr, \Probabilistic intra-retinal layer segmentation in 3-D OCT images using global shape regularization," Med. Image Anal., vol. 18, no. 5, pp. 781-794, 2014.

Analele Universitatii "Ovidius" Constanta - Seria Matematica

The Journal of "Ovidius" University of Constanta

Journal Information

IMPACT FACTOR 2017: 0.452
5-year IMPACT FACTOR: 0.512

CiteScore 2017: 0.59

SCImago Journal Rank (SJR) 2017: 0.316
Source Normalized Impact per Paper (SNIP) 2017: 0.790

Mathematical Citation Quotient (MCQ) 2016: 0.10

Target Group

researchers in all fields of pure and applied mathematics


All Time Past Year Past 30 Days
Abstract Views 0 0 0
Full Text Views 234 202 17
PDF Downloads 78 70 5