Annals of Statistics

Limit distribution theory for maximum likelihood estimation of a log-concave density

Fadoua Balabdaoui, Kaspar Rufibach, and Jon A. Wellner

Full-text: Open access


We find limiting distributions of the nonparametric maximum likelihood estimator (MLE) of a log-concave density, that is, a density of the form f0=exp ϕ0 where ϕ0 is a concave function on ℝ. The pointwise limiting distributions depend on the second and third derivatives at 0 of Hk, the “lower invelope” of an integrated Brownian motion process minus a drift term depending on the number of vanishing derivatives of ϕ0=log f0 at the point of interest. We also establish the limiting distribution of the resulting estimator of the mode M(f0) and establish a new local asymptotic minimax lower bound which shows the optimality of our mode estimator in terms of both rate of convergence and dependence of constants on population values.

Article information

Ann. Statist., Volume 37, Number 3 (2009), 1299-1331.

First available in Project Euclid: 10 April 2009

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 62N01: Censored data models 62G20: Asymptotic properties
Secondary: 62G05: Estimation

Asymptotic distribution integral of Brownian motion invelope process log-concave density estimation lower bounds maximum likelihood mode estimation nonparametric estimation qualitative assumptions shape constraints strongly unimodal unimodal


Balabdaoui, Fadoua; Rufibach, Kaspar; Wellner, Jon A. Limit distribution theory for maximum likelihood estimation of a log-concave density. Ann. Statist. 37 (2009), no. 3, 1299--1331. doi:10.1214/08-AOS609.

Export citation


  • An, M. Y. (1998). Logconcavity versus logconvexity: A complete characterization. J. Econom. Theory 80 350–369.
  • Anevski, D. and Hössjer, O. (2006). A general asymptotic scheme for inference under order restrictions. Ann. Statist. 34 1874–1930.
  • Bagnoli, M. and Bergstrom, T. (2005). Log-concave probability and its applications. Econom. Theory 26 445–469.
  • Balabdaoui, F. and Wellner, J. A. (2007). Estimation of a k-monotone density: Limiting distribution theory and the spline connection. Ann. Statist. 35 2536–2564.
  • Barlow, R. E. and Proschan, F. (1975). Statistical Theory of Reliability and Life Testing. Holt, Rinehart and Winston, New York.
  • Birgé, L. (1989). The Grenander estimator: A nonasymptotic approach. Ann. Statist. 17 1532–1549.
  • Birgé, L. (1997). Estimation of unimodal densities without smoothness assumptions. Ann. Statist. 25 970–981.
  • Brunner, L. J. and Lo, A. Y. (1989). Bayes methods for a symmetric unimodal density and its mode. Ann. Statist. 17 1550–1566.
  • Chang, G. and Walther, G. (2007). Clustering with mixtures of log-concave distributions. Comput. Statist. Data Anal. 51 6242–6251.
  • Chernoff, H. (1964). Estimation of the mode. Ann. Inst. Statist. Math. 16 31–41.
  • Cule, M., Gramacy, R. and Samworth, R. (2007). LogConcDEAD: Maximum likelihood estimation of a log-concave density. R package version 1.0-5.
  • Dalenius, T. (1965). The mode—a neglected statistical parameter. J. Roy. Statist. Soc. Ser. A 128 110–117.
  • Dharmadhikari, S. and Joag-Dev, K. (1988). Unimodality, Convexity, and Applications. Academic Press, Boston, MA.
  • Dümbgen, L., Hüsler, A. and Rufibach, K. (2007). Active set and EM algorithms for log-concave densities based on complete and censored data. Technical report, Univ. Bern. Available at arXiv:0707.4643.
  • 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. To appear. Available at arXiv:0709.0334.
  • Eddy, W. F. (1980). Optimum kernel estimators of the mode. Ann. Statist. 8 870–882.
  • Eddy, W. F. (1982). The asymptotic distributions of kernel estimators of the mode. Z. Wahrsch. Verw. Gebiete 59 279–290.
  • Ekblom, H. (1972). A Monte Carlo investigation of mode estimators in small samples. Appl. Statist. 21 177–184.
  • Grenander, U. (1956). On the theory of mortality measurement. II. Skand. Aktuarietidskr. 39 125–153 (1957).
  • Grenander, U. (1965). Some direct estimates of the mode. Ann. Math. Statist. 36 131–138.
  • Groeneboom, P. (1996). Lectures on inverse problems. In Lectures on Probability Theory and Statistics (Saint-Flour, 1994). Lecture Notes in Math. 1648 67–164. Springer, Berlin.
  • Groeneboom, P., Jongbloed, G. and Wellner, J. A. (2001a). A canonical process for estimation of convex functions: The “invelope” of integrated Brownian motion +t4. 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.
  • Groeneboom, P., Maathuis, M. H. and Wellner, J. A. (2008). Current status data with competing risks: Limiting distribution of the MLE. Ann. Statist. 36 1064–1089.
  • Grund, B. and Hall, P. (1995). On the minimisation of Lp error in mode estimation. Ann. Statist. 23 2264–2284.
  • Hall, P. (1982). Asymptotic theory of Grenander’s mode estimator. Z. Wahrsch. Verw. Gebiete 60 315–334.
  • Has’minskiĭ, R. Z. (1979). Lower bound for the risks of nonparametric estimates of the mode. In Contributions to Statistics 91–97. Reidel, Dordrecht.
  • Herrmann, E. and Ziegler, K. (2004). Rates on consistency for nonparametric estimation of the mode in absence of smoothness assumptions. Statist. Probab. Lett. 68 359–368.
  • Ho, M.-W. (2006a). Bayes estimation of a symmetric unimodal density via S-paths. J. Comput. Graph. Statist. 15 848–860.
  • Ho, M.-W. (2006b). Bayes estimation of a unimodal density via S-paths. Technical report, National University of Singapore. Available at
  • Ibragimov, I. A. (1956). On the composition of unimodal distributions. Teor. Veroyatnost. i Primenen. 1 283–288.
  • Jankowski, H. and Wellner, J. A. (2007). Nonparametric estimation of a convex bathtub-shaped hazard function. Technical report, Univ. Washington.
  • Jongbloed, G. (1995). Three statistical inverse problems. Ph.D. thesis, Delft Univ. Technology.
  • Jongbloed, G. (2000). Minimax lower bounds and moduli of continuity. Statist. Probab. Lett. 50 279–284.
  • Kim, J. and Pollard, D. (1990). Cube root asymptotics. Ann. Statist. 18 191–219.
  • Klemelä, J. (2005). Adaptive estimation of the mode of a multivariate density. J. Nonparametr. Statist. 17 83–105.
  • Lepskiĭ, O. V. (1991). Asymptotically minimax adaptive estimation. I. Upper bounds. Optimally adaptive estimates. Teor. Veroyatnost. i Primenen. 36 645–659.
  • Lepskiĭ, O. V. (1992). Asymptotically minimax adaptive estimation. II. Schemes without optimal adaptation. Adaptive estimates. Teor. Veroyatnost. i Primenen. 37 468–481.
  • Leurgans, S. (1982). Asymptotic distributions of slope-of-greatest-convex-minorant estimators. Ann. Statist. 10 287–296.
  • Marshall, A. W. and Olkin, I. (1979). Inequalities: Theory of Majorization and Its Applications. Mathematics in Science and Engineering 143. Academic Press [Harcourt Brace Jovanovich Publishers], New York.
  • Marshall, A. W. and Olkin, I. (2007). Life Distributions. Springer, New York.
  • Meyer, M. C. (2001). An alternative unimodal density estimator with a consistent estimate of the mode. Statist. Sinica 11 1159–1174.
  • Meyer, M. C. and Woodroofe, M. (2004). Consistent maximum likelihood estimation of a unimodal density using shape restrictions. Canad. J. Statist. 32 85–100.
  • Müller, H.-G. (1989). Adaptive nonparametric peak estimation. Ann. Statist. 17 1053–1069.
  • Müller, S. and Rufibach, K. (2009). Smoothed tail index estimation. J. Statist. Comput. Simulation. To appear. Available at arXiv:math/0612140.
  • Pal, J. K., Woodroofe, M. B. and Meyer, M. C. (2007). Estimating a Polya frequency function. In Complex Datasets and Inverse Problems: Tomography, Networks, and Beyond. IMS Lecture Notes-Monograph Series 54 239–249.
  • Parzen, E. (1962). On estimation of a probability density function and mode. Ann. Math. Statist. 33 1065–1076.
  • Prakasa Rao, B. L. S. (1969). Estimation of a unimodal density. Sankhyā Ser. A 31 23–36.
  • Romano, J. P. (1988). On weak convergence and optimality of kernel density estimates of the mode. Ann. Statist. 16 629–647.
  • Rufibach, K. (2006). Log-concave density estimation and bump hunting for I.I.D. observations. Ph.D. thesis, Univ. Bern and Göttingen.
  • Rufibach, K. (2007). Computing maximum likelihood estimators of a log-concave density function. J. Statist. Comput. Simulation 77 561–574.
  • Rufibach, K. and Dümbgen, L. (2007). Logcondens: estimate a log-concave probability density from i.i.d. observations. R package version 1.3.1.
  • van der Vaart, A. W. and Wellner, J. A. (1996). Weak Convergence and Empirical Processes with Applications in Statistics. Springer, New York.
  • Venter, J. H. (1967). On estimation of the mode. Ann. Math. Statist 38 1446–1455.
  • Vieu, P. (1996). A note on density mode estimation. Statist. Probab. Lett. 26 297–307.
  • Walther, G. (2002). Detecting the presence of mixing with multiscale maximum likelihood. J. Amer. Statist. Assoc. 97 508–513.
  • Wegman, E. J. (1970a). Maximum likelihood estimation of a unimodal density function. Ann. Math. Statist. 41 457–471.
  • Wegman, E. J. (1970b). Maximum likelihood estimation of a unimodal density. II. Ann. Math. Statist. 41 2169–2174.
  • Wegman, E. J. (1971). A note on the estimation of the mode. Ann. Math. Statist. 42 1909–1915.
  • Wright, F. T. (1981). The asymptotic behavior of monotone regression estimates. Ann. Statist. 9 443–448.
  • Ziegler, K. (2004). Adaptive kernel estimation of the mode in a nonparametric random design regression model. Probab. Math. Statist. 24 213–235.