The Annals of Statistics

Global rates of convergence of the MLEs of log-concave and $s$-concave densities

Charles R. Doss and Jon A. Wellner

Full-text: Open access


We establish global rates of convergence for the Maximum Likelihood Estimators (MLEs) of log-concave and $s$-concave densities on $\mathbb{R}$. The main finding is that the rate of convergence of the MLE in the Hellinger metric is no worse than $n^{-2/5}$ when $-1<s<\infty$ where $s=0$ corresponds to the log-concave case. We also show that the MLE does not exist for the classes of $s$-concave densities with $s<-1$.

Article information

Ann. Statist., Volume 44, Number 3 (2016), 954-981.

Received: June 2013
Revised: September 2015
First available in Project Euclid: 11 April 2016

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 62G07: Density estimation
Secondary: 62G05: Estimation 62G20: Asymptotic properties

Bracketing entropy consistency empirical processes global rate Hellinger metric log-concave $s$-concave


Doss, Charles R.; Wellner, Jon A. Global rates of convergence of the MLEs of log-concave and $s$-concave densities. Ann. Statist. 44 (2016), no. 3, 954--981. doi:10.1214/15-AOS1394.

Export citation


  • [1] Adler, R. J., Feldman, R. E. and Taqqu, M. S., eds. (1998). A Practical Guide to Heavy Tails. Birkhäuser, Boston, MA.
  • [2] 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.
  • [3] Birgé, L. (1997). Estimation of unimodal densities without smoothness assumptions. Ann. Statist. 25 970–981.
  • [4] Birgé, L. and Massart, P. (1993). Rates of convergence for minimum contrast estimators. Probab. Theory Related Fields 97 113–150.
  • [5] Borell, C. (1974). Convex measures on locally convex spaces. Ark. Mat. 12 239–252.
  • [6] Borell, C. (1975). Convex set functions in $d$-space. Period. Math. Hungar. 6 111–136.
  • [7] Brascamp, H. J. and Lieb, E. H. (1976). On extensions of the Brunn–Minkowski and Prékopa–Leindler theorems, including inequalities for log concave functions, and with an application to the diffusion equation. J. Funct. Anal. 22 366–389.
  • [8] Bronšteĭn, E. M. (1976). $\epsilon $-entropy of convex sets and functions. Sibirsk. Mat. Zh. 17 508–514, 715.
  • [9] 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.
  • [10] 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.
  • [11] Dharmadhikari, S. and Joag-Dev, K. (1988). Unimodality, Convexity, and Applications. Academic Press, Boston, MA.
  • [12] Doss, C. R. and Wellner, J. A. (2015). Supplement to “Global rates of convergence of the MLEs of log-concave and $s$-concave densities.” DOI:10.1214/15-AOS1394SUPP.
  • [13] Doss, C. R. (2013). Shape-constrained inference for concave-transformed densities and their modes. Ph.D. thesis, Dept. Statistics, Univ. Washington, Seattle, WA.
  • [14] Doss, C. R. and Wellner, J. A. (2013). Global rates of convergence of the MLEs of log-concave and $s$-concave densities. Available at arXiv:1306.1438.
  • [15] Doss, C. R. and Wellner, J. A. (2015). Inference for the mode of a log-concave density. Technical report, Univ. Washington, Seattle, WA.
  • [16] Dryanov, D. (2009). Kolmogorov entropy for classes of convex functions. Constr. Approx. 30 137–153.
  • [17] Dudley, R. M. (1984). A course on empirical processes. In École D’été de Probabilités de Saint-Flour, XII—1982. Lecture Notes in Math. 1097 1–142. Springer, Berlin.
  • [18] Dudley, R. M. (1999). Uniform Central Limit Theorems. Cambridge Studies in Advanced Mathematics 63. Cambridge Univ. Press, Cambridge.
  • [19] 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.
  • [20] Dümbgen, L., Samworth, R. and Schuhmacher, D. (2011). Approximation by log-concave distributions, with applications to regression. Ann. Statist. 39 702–730.
  • [21] Guntuboyina, A. (2012). Optimal rates of convergence for convex set estimation from support functions. Ann. Statist. 40 385–411.
  • [22] Guntuboyina, A. and Sen, B. (2013). Covering numbers for convex functions. IEEE Trans. Inform. Theory 59 1957–1965.
  • [23] Guntuboyina, A. and Sen, B. (2015). Global risk bounds and adaptation in univariate convex regression. Probab. Theory Related Fields 163 379–411.
  • [24] Han, Q. and Wellner, J. A. (2016). Approximation and estimation of $s$-concave densities via Rényi divergences. Ann. Statist. 44 1332–1359.
  • [25] Kim, A. K. H. and Samworth, R. J. (2015). Global rates of convergence in log-concave density estimation. Available at arXiv:1404.2298v2.
  • [26] Koenker, R. and Mizera, I. (2010). Quasi-concave density estimation. Ann. Statist. 38 2998–3027.
  • [27] Pal, J. K., Woodroofe, M. and Meyer, M. (2007). Estimating a Polya frequency function${}_{2}$. In Complex Datasets and Inverse Problems. Institute of Mathematical Statistics Lecture Notes—Monograph Series 54 239–249. IMS, Beachwood, OH.
  • [28] Prékopa, A. (1995). Stochastic Programming. Mathematics and Its Applications 324. Kluwer Academic, Dordrecht.
  • [29] Resnick, S. I. (2007). Heavy-Tail Phenomena: Probabilistic and Statistical Modeling. Springer, New York.
  • [30] Rinott, Y. (1976). On convexity of measures. Ann. Probab. 4 1020–1026.
  • [31] Rockafellar, R. T. (1970). Convex Analysis. Princeton Mathematical Series 28. Princeton Univ. Press, Princeton, NJ.
  • [32] Schuhmacher, D., Hüsler, A. and Dümbgen, L. (2011). Multivariate log-concave distributions as a nearly parametric model. Stat. Risk Model. 28 277–295.
  • [33] Seregin, A. and Wellner, J. A. (2010). Nonparametric estimation of multivariate convex-transformed densities. Ann. Statist. 38 3751–3781.
  • [34] van de Geer, S. (1993). Hellinger-consistency of certain nonparametric maximum likelihood estimators. Ann. Statist. 21 14–44.
  • [35] van de Geer, S. A. (2000). Applications of Empirical Process Theory. Cambridge Series in Statistical and Probabilistic Mathematics 6. Cambridge Univ. Press, Cambridge.
  • [36] van der Vaart, A. W. and Wellner, J. A. (1996). Weak Convergence and Empirical Processes. Springer, New York.
  • [37] Walther, G. (2002). Detecting the presence of mixing with multiscale maximum likelihood. J. Amer. Statist. Assoc. 97 508–513.
  • [38] Wong, W. H. and Shen, X. (1995). Probability inequalities for likelihood ratios and convergence rates of sieve MLEs. Ann. Statist. 23 339–362.

Supplemental materials