Theoretical properties of the log-concave maximum likelihood estimator of a multidimensional density

Electronic Journal of Statistics

Volumn 4, Issue , 2010, Pages 254-270

  Cule, Madeleine ^a   Samworth, Richard ^a  

^a UNIVERSITY OF CAMBRIDGE   (United Kingdom)

Author keywords

Consistency; Kullback Leibler divergence; Log concavity; Maximum likelihood estimation; Model misspecification

Indexed keywords

EID: 77957584324     PISSN: 19357524     EISSN: None     Source Type: Journal    
DOI: 10.1214/09-EJS505     Document Type: Article

References (34)

1. Bagnoli, M. And Bergstrom, T. (1989), Log-concave probability and its applications, Unpublished manuscript, University of Michigan.
2. Balabdaoui, F., Jankowski, H. K., Pavlides, M., Seregin, A. And Wellner, J. A. (2010), On the Grenander estimator at zero, Statistica Sinica, to appear.
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.
4. Balabdaoui, F. And Wellner, J. A. (2007), Estimation of a k-monotone density: limit distribution theory and the spline connection, Ann. Statist., 35, 2536–2564.
5. Balabdaoui, F. And Wellner, J. A. (2010), Estimation of a k-monotone density: characterizations, consistency and minimax lower bounds, Statistica Neerlandica, 64, 45–70.
6. Billingsley, P. (1999), Convergence of Probability Measures, Wiley, New York.
7. Birgé, L. (1989), The Grenander estimator: a non-asymptotic approach, Ann. Statist., 17, 1532–1549.
8. Birgé, L. (1997), Estimation of unimodal densities without smoothness assumptions, Ann. Statist., 25, 970–981.
9. Bhattacharya, R. N. And Rao, R. R. Normal Approximation and Asymptotic Expansions, Wiley, New York.
10. Cule, M. L., Gramacy, R. B. And Samworth, R. J. (2007), LogConcDEAD: an R package for log-concave density estimation in arbitrary dimensions. Available from http://cran.r-project.org/.
11. Cule, M. L., Gramacy, R. B. And Samworth, R. J. (2009) LogConcDEAD: an R package for maximum likelihood estimation of a multivariate log-concave density, J. Statist. Software, 29, Issue 2.
12. Cule, M. L., Samworth, R. J. And Stewart, M. I. (2010), Maximum likelihood estimation of a multidimensional log-concave density, J. Roy. Statist. Soc., Ser. B (with discussion), to appear.
13. Devroye, L. (1987), A course in density estimation, Birkhäuser Boston Inc., Boston MA.
14. Dharmadhikari, S. And Joag-Dev, K. (1988), Unimodality, Convexity and Applications, Academic Press, Inc., San Diego, CA.
15. Dümbgen, L., Hüsler, A. And Rufibach, K. (2007), Active Set and EM Algorithms for Log-Concave Densities Based on Complete and Censored Data. Preprint. http://arxiv.org/abs/0707.4643.
16. 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.
17. Dümbgen, L., Rufibach, K. And Wellner, J. A. (2007), Marshall’s lemma for convex density estimation, Asymptotics: particles, processes and inverse problems, 101–107, IMS Lecture Notes Monogr. Ser., 55, Inst. Math. Statist., Beachwood, OH.
18. Folland, G. B. (1999), Real Analysis, Wiley, New York.
19. Grenander, U. (1956), On the theory of mortality measurement. II. Skand. Aktuarietidskr., 39, 125–153.
20. Groeneboom, P., Jongbloed, G. And Wellner, J. A. (2001), Estimation of a convex function: Characterization and asymptotic theory, Ann. Statist., 29, 1653–1698.
21. Marshall, A. W. And Proschan, F. (1965), Maximum likelihood estimation for distributions with monotone failure rate, Ann. Math. Statist., 36, 69–77.
22. Pal, J., Woodroofe, M. And Meyer, M. (2007), Estimating a Polya frequency function. In Complex datasets and Inverse problems: Tomography, Networks and Beyond (eds. R. Liu, W. Strawderman, C.-H. Zhang), pp. 239–249. IMS Lecture Notes - Monograph Series 54.
23. Prakasa Rao, B. L. S. (1969), Estimation of a unimodal density, Sankhya, Ser. A., 31, 23–36.
24. Rao, M. M. And Ren, Z. D. (1991), Theory of Orlicz spaces, CRC Press, New York.
25. Rockafellar, R. T. (1997), Convex Analysis, Princeton Univ. Press, Princeton, New Jersey.
26. Rufibach, K. (2007), Computing Maximum Likelihood Estimators of a logconcave Density Function, J. Stat. Comput. Simul., 77, 561–574.
27. Schuhmacher, D. And Dümbgen, L. (2010), Consistency of Multivariate Log-Concave Density Estimators, Statist. Probab. Lett., 80, 376–380.
28. Schuhmacher, D., Hüsler, A. And Dümbgen, L. (2009), Multivariate Log-Concave Distributions as a Nearly Parametric Model. Preprint. http://arxiv.org/pdf/0907.0250.
29. Seregin, A. And Wellner, J. A. (2009), Nonparametric estimation of multivariate convex-transformed densities. Preprint. http://uk.arxiv.org/pdf/0911.4151.pdf.
30. van de Geer, S. (2000), Empirical Processes in M-estimation, Cambridge Univ. Press, Cambridge.
31. van de Geer, S. (1993), Hellinger-consistency of certain nonparametric maximum likelihood estimators, Ann. Statist., 21, 14–44.
32. van der Vaart, A. W. And Wellner, J. A. (1996), Weak Convergence and Empirical Processes, Springer-Verlag, New York.
33. Walther, G. (2002), Detecting the presence of mixing with multiscale maximum likelihood, J. Amer. Statist. Assoc., 97, 508–513.
34. Walther, G. (2010), Inference and modeling with log-concave distributions, Statistical Science, to appear.