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: Access has been disabled (more information)

Abstract

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

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

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

Permanent link to this document
http://projecteuclid.org/euclid.aos/1460381683

Digital Object Identifier
doi:10.1214/15-AOS1394

Mathematical Reviews number (MathSciNet)
MR3485950

Zentralblatt MATH identifier
1338.62101

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

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

Citation

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. http://projecteuclid.org/euclid.aos/1460381683.


Export citation

References

  • [1] Adler, R. J., Feldman, R. E. and Taqqu, M. S., eds. (1998). A Practical Guide to Heavy Tails. Birkhäuser, Boston, MA.
    Mathematical Reviews (MathSciNet): MR1652283
  • [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.
    Mathematical Reviews (MathSciNet): MR2509075
    Digital Object Identifier: doi:10.1214/08-AOS609
    Project Euclid: euclid.aos/1239369023
  • [3] Birgé, L. (1997). Estimation of unimodal densities without smoothness assumptions. Ann. Statist. 25 970–981.
    Mathematical Reviews (MathSciNet): MR1447736
    Digital Object Identifier: doi:10.1214/aos/1069362733
    Project Euclid: euclid.aos/1069362733
  • [4] Birgé, L. and Massart, P. (1993). Rates of convergence for minimum contrast estimators. Probab. Theory Related Fields 97 113–150.
    Mathematical Reviews (MathSciNet): MR1240719
    Digital Object Identifier: doi:10.1007/BF01199316
  • [5] Borell, C. (1974). Convex measures on locally convex spaces. Ark. Mat. 12 239–252.
    Mathematical Reviews (MathSciNet): MR388475
  • [6] Borell, C. (1975). Convex set functions in $d$-space. Period. Math. Hungar. 6 111–136.
    Mathematical Reviews (MathSciNet): MR404559
    Digital Object Identifier: doi:10.1007/BF02018814
  • [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.
    Mathematical Reviews (MathSciNet): MR450480
    Digital Object Identifier: doi:10.1016/0022-1236(76)90004-5
  • [8] Bronšteĭn, E. M. (1976). $\epsilon $-entropy of convex sets and functions. Sibirsk. Mat. Zh. 17 508–514, 715.
    Mathematical Reviews (MathSciNet): MR415155
  • [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.
    Mathematical Reviews (MathSciNet): MR2645484
    Digital Object Identifier: doi:10.1214/09-EJS505
    Project Euclid: euclid.ejs/1266416850
  • [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.
    Mathematical Reviews (MathSciNet): MR2758237
    Digital Object Identifier: doi:10.1111/j.1467-9868.2010.00753.x
  • [11] Dharmadhikari, S. and Joag-Dev, K. (1988). Unimodality, Convexity, and Applications. Academic Press, Boston, MA.
    Mathematical Reviews (MathSciNet): MR954608
  • [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.
    Mathematical Reviews (MathSciNet): MR3485950
    Digital Object Identifier: doi:10.1214/15-AOS1394
    Project Euclid: euclid.aos/1460381683
  • [13] Doss, C. R. (2013). Shape-constrained inference for concave-transformed densities and their modes. Ph.D. thesis, Dept. Statistics, Univ. Washington, Seattle, WA.
    Mathematical Reviews (MathSciNet): MR3193124
  • [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.
    arXiv: arXiv:1306.1438
    Mathematical Reviews (MathSciNet): MR3485950
    Digital Object Identifier: doi:10.1214/15-AOS1394
    Project Euclid: euclid.aos/1460381683
  • [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.
    Mathematical Reviews (MathSciNet): MR2519658
    Digital Object Identifier: doi:10.1007/s00365-009-9053-3
  • [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.
    Mathematical Reviews (MathSciNet): MR876079
  • [18] Dudley, R. M. (1999). Uniform Central Limit Theorems. Cambridge Studies in Advanced Mathematics 63. Cambridge Univ. Press, Cambridge.
    Mathematical Reviews (MathSciNet): MR1720712
  • [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.
    Mathematical Reviews (MathSciNet): MR2546798
    Digital Object Identifier: doi:10.3150/08-BEJ141
    Project Euclid: euclid.bj/1233669882
  • [20] Dümbgen, L., Samworth, R. and Schuhmacher, D. (2011). Approximation by log-concave distributions, with applications to regression. Ann. Statist. 39 702–730.
    Mathematical Reviews (MathSciNet): MR2816336
    Digital Object Identifier: doi:10.1214/10-AOS853
    Project Euclid: euclid.aos/1299680952
  • [21] Guntuboyina, A. (2012). Optimal rates of convergence for convex set estimation from support functions. Ann. Statist. 40 385–411.
    Mathematical Reviews (MathSciNet): MR3014311
    Digital Object Identifier: doi:10.1214/11-AOS959
    Project Euclid: euclid.aos/1334581747
  • [22] Guntuboyina, A. and Sen, B. (2013). Covering numbers for convex functions. IEEE Trans. Inform. Theory 59 1957–1965.
    Mathematical Reviews (MathSciNet): MR3043776
    Digital Object Identifier: doi:10.1109/TIT.2012.2235172
  • [23] Guntuboyina, A. and Sen, B. (2015). Global risk bounds and adaptation in univariate convex regression. Probab. Theory Related Fields 163 379–411.
    Mathematical Reviews (MathSciNet): MR3405621
    Digital Object Identifier: doi:10.1007/s00440-014-0595-3
  • [24] Han, Q. and Wellner, J. A. (2016). Approximation and estimation of $s$-concave densities via Rényi divergences. Ann. Statist. 44 1332–1359.
    Mathematical Reviews (MathSciNet): MR3485962
    Digital Object Identifier: doi:10.1214/15-AOS1408
    Project Euclid: euclid.aos/1460381695
  • [25] Kim, A. K. H. and Samworth, R. J. (2015). Global rates of convergence in log-concave density estimation. Available at arXiv:1404.2298v2.
    arXiv: arXiv:1404.2298v2
  • [26] Koenker, R. and Mizera, I. (2010). Quasi-concave density estimation. Ann. Statist. 38 2998–3027.
    Mathematical Reviews (MathSciNet): MR2722462
    Digital Object Identifier: doi:10.1214/10-AOS814
    Project Euclid: euclid.aos/1282315406
  • [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.
    Mathematical Reviews (MathSciNet): MR2459192
    Digital Object Identifier: doi:10.1214/074921707000000184
  • [28] Prékopa, A. (1995). Stochastic Programming. Mathematics and Its Applications 324. Kluwer Academic, Dordrecht.
    Mathematical Reviews (MathSciNet): MR1375234
  • [29] Resnick, S. I. (2007). Heavy-Tail Phenomena: Probabilistic and Statistical Modeling. Springer, New York.
    Mathematical Reviews (MathSciNet): MR2271424
  • [30] Rinott, Y. (1976). On convexity of measures. Ann. Probab. 4 1020–1026.
    Mathematical Reviews (MathSciNet): MR428540
    Digital Object Identifier: doi:10.1214/aop/1176995947
  • [31] Rockafellar, R. T. (1970). Convex Analysis. Princeton Mathematical Series 28. Princeton Univ. Press, Princeton, NJ.
    Mathematical Reviews (MathSciNet): MR274683
  • [32] Schuhmacher, D., Hüsler, A. and Dümbgen, L. (2011). Multivariate log-concave distributions as a nearly parametric model. Stat. Risk Model. 28 277–295.
    Mathematical Reviews (MathSciNet): MR2838319
  • [33] Seregin, A. and Wellner, J. A. (2010). Nonparametric estimation of multivariate convex-transformed densities. Ann. Statist. 38 3751–3781.
    Mathematical Reviews (MathSciNet): MR2766867
    Digital Object Identifier: doi:10.1214/10-AOS840
    Project Euclid: euclid.aos/1291126972
  • [34] van de Geer, S. (1993). Hellinger-consistency of certain nonparametric maximum likelihood estimators. Ann. Statist. 21 14–44.
    Mathematical Reviews (MathSciNet): MR1212164
    Digital Object Identifier: doi:10.1214/aos/1176349013
    Project Euclid: euclid.aos/1176349013
  • [35] van de Geer, S. A. (2000). Applications of Empirical Process Theory. Cambridge Series in Statistical and Probabilistic Mathematics 6. Cambridge Univ. Press, Cambridge.
    Mathematical Reviews (MathSciNet): MR1739079
  • [36] van der Vaart, A. W. and Wellner, J. A. (1996). Weak Convergence and Empirical Processes. Springer, New York.
    Mathematical Reviews (MathSciNet): MR1385671
  • [37] Walther, G. (2002). Detecting the presence of mixing with multiscale maximum likelihood. J. Amer. Statist. Assoc. 97 508–513.
    Mathematical Reviews (MathSciNet): MR1941467
    Digital Object Identifier: doi:10.1198/016214502760047032
  • [38] Wong, W. H. and Shen, X. (1995). Probability inequalities for likelihood ratios and convergence rates of sieve MLEs. Ann. Statist. 23 339–362.
    Mathematical Reviews (MathSciNet): MR1332570
    Digital Object Identifier: doi:10.1214/aos/1176324524
    Project Euclid: euclid.aos/1176324524

Supplemental materials

  • Technical proofs. In the supplement, we provide additional proofs and technical details that were omitted from the main paper.
    Digital Object Identifier: doi:10.1214/15-AOS1394SUPP
    Supplemental files available for subscribers.

The Institute of Mathematical Statistics

The Annals of Statistics

New content alerts

Email RSS ToC RSS Article
  • You have access to this content.
  • You have partial access to this content.
  • You do not have access to this content.
Turn Off MathJax

What is MathJax?

More like this

+ See more