Estimation of a κ-monotone density: Characterizations, consistency and minimax lower bounds

Statistica Neerlandica

Volumn 64, Issue 1, 2010, Pages 45-70

  Balabdaoui, Fadoua ^a   Wellner, Jon A ^b  

^a UNIVERSITÉ PARIS DAUPHINE   (France)

^b UNIVERSITY OF WASHINGTON   (United States)

Author keywords

Completely monotone; Least squares; Maximum likelihood; Minimax risk; Mixture models; Multiply monotone; Non parametric estimation; Rates of convergence; Shape constraints

Indexed keywords

EID: 76149113596     PISSN: 00390402     EISSN: 14679574     Source Type: Journal    
DOI: 10.1111/j.1467-9574.2009.00438.x     Document Type: Article

References (42)

1. Anevski, D 1994 Estimating the derivative of a convex density, Technical Report 1994:8, Department of Mathematical Statistics, University of Lund.
2. Anevski, D. (2003 Estimating the derivative of a convex density, Statistica Neerlandica 57, 245 257.
3. Apostol, T. M. (1957 Mathematical analysis: a modern approach to advanced calculus, Addision-Wesley Publishing Company, Inc., Reading, MA.
4. Ayer M., H. D. Brunk, G. M. Ewing, W. T. Reid E. Silverman (1955 An empirical distribution function for sampling with incomplete information, The Annals of Mathematcial Statistics 26, 641 647.
5. Balabdaoui, F. (2007 Consistent estimation of a convex density at the origin, Mathematical Methods of Statistics 16, 77 95.
6. Balabdaoui, F. J. A. Wellner (2004 Estimation of a K-monotone density, part 2: algorithm for computation and numerical results, Technical Report 460, Department of Statistics, University of Washington.
7. Balabdaoui, F. J. A. Wellner (2007 Estimation of a K-monotone density: Limit distribution theory and the spline connection, The Annals of Statistics 35, 2536 2564.
8. Brunk H. D. (1981 Probability theory and elements of measure theory, Academic Press Inc. [Harcourt Brace Jovanovich Publisher], London. Second edition of the translation by R. B. Burckel from the third German edition, Probability and Mathematical Statistics.
9. Brunk H. D. (1958 On the estimation of parameters restricted by inequalities. Annals of Mathematical Statistics 29, 437 454.
10. De Boor, C. (1978 A practical guide to splines, vol. 27 of Applied Mathematical Science, Springer-Verlag, New York.
11. DeVore, R. A. G. G. Lorentz (1993 Constructive approximation, vol. 303 of Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], Springer-Verlag Berlin.
12. Donoho, D. L. R. C. Liu (1987 Geometrizing rates of convergence. i, Technical Report. 137, Department of Statistics, University of Califonia, Berkeley.
13. Donoho, D. L. R. C. Liu (1991 Geometrizing rates of convergence. II III, The Annals of Statistics 19, 633 667, 668 701.
14. van Eeden, C 1957a Maximum likelihood estimation of partially or completely ordered parameters. I, Koninklijke Nederlandse Akademie van Wetenschappen Proceedings. Series A Mathematical Sciences 19, 128 136.
15. van Eeden, C. (1957b Maximum likelihood estimation of partially or completely ordered parameters. II, Koninklijke Nederlandse Akademie van Wetenschappen Proceedings. Series A Mathematical Sciences 19, 201 211.
16. Feller, W. (1971 An introduction to probability theory and its applications, vol. II, 2nd edition, John Wiley & Sons Inc., New york.
17. van de Geer, S. (1993 Hellinger-consistency of certain nonparametric maximum likelihood estimators, The Annals of Statistics 21, 14 44.
18. Gneiting, T. (1998 On the Bernstein-Hausdorff-Widder conditions for completely monotone function. Expositionces Mathaticae 16, 88 119.
19. Gneiting, T. (1999 Radial positive definite functions generated by Euclid's hat, Journal of Multivariate Analysis 69, 88 119.
20. Grenander, U. (1956a On the theory of mortality measurement. I Skandinavisk Aktuarietidskrift 39, 70 96.
21. Grenander, U. (1956b On the theory of mortality measurement. II, Skandinavisk Aktuarietidskrift 39, 125 153.
22. Groeneboom, P 1985 Estimating a monotone density, in : Proceedings of the Berkeley conference in honor of Jerzy Neyman and Jack Kiefer Vol. II (Berkeley, CA. 1983 Wadsworth Statist./Probab. Ser., Wadsworth, Belmont, CA.
23. Groeneboom, P. (1989 Brownian motion with a parabolic drift and Airy functions, Probability Theory and Related Fields 81, 79 109.
24. Groeneboom, P. (1996 Lectures on inverse problems, in : Lectures on probability theory and statistics (Saint-Flour 1994), vol. 1648 of Lecture Notes in Mathematics, Springer, Berlin, pp. 67 164.
25. Groeneboom, P., G. Jongbloed J. A. Wellner (2001 Estimation of a convex function: characterizations and asymptotic theory, The Annals of Statistics 29, 1653 1698.
26. Groeneboom, P., G. Jongbloed J. A. Wellner (2008 The support reduction algorithm for computing non-parametric function estimates in mixture models, Scandinavian Journal of Statistics 35, 385 399.
27. Hall, W. J. J. A. Wellner (1979 The rate of convergence in law of the maximum of an exponential sample, Statistica Neerlandica 33, 151 154.
28. Hampel, F. R. (1987). Design modelling and anlysis of some biological datasets, in : design, data and analysis, by some friends of Cuthbert Daniel C. Mallows (ed Wiley, New york, pp. 111 115.
29. Jewell, N. P. (1982 Mixtures of exponential distributions, The Annals of Statistics 10, 479 484.
30. Jongbloed, G. (1995 Three statistical inverse problems, Ph.D. Thesis, Delft University of Technology.
31. Jongbloed, G. (2000 Minimax lower bounds and moduli of continuity, Statistics & Probability Letters 50, 279 284.
32. Kim, J. D. Pollard (1990 Cube root asymptotics, The Annals of Statistics 18, 191 219.
33. Lévy, P. (1962) Extensions d'un théorème de D. Dugué et M, Girault. Probability Theory and Related Fields 1, 159 173.
34. Lindsay, B. G. (1983 The geometry of mixture likelihoods: a general theory, The Annals of Statistics 11, 86 94.
35. Pfanzagl, J. (1988 Consistency of maximum likelihood estimators for certain nonparametric families in particular: mixtures, Journal of Statistical Planning and Inference 19, 137 158.
36. Prakasa Rao, B. L. S. (1969 Estimation of a unimodal density, Sankhy Series A 31, 23 36.
37. Silverman, B. W. (1982 On the estimation of a probability density function by the maximum penalized likelihood method, The Annals of Statistics 10, 795 810.
38. van der vaart, A. J. A. Wellner (2000 Preservation theorems for Glivenko-Cantelli and unifrom Gilvenko-Cantelli classes, in : Hign dimensional probability, II (Seattle, WA, 1999), vol. 47 of Progr. Probab., BirKhäuser Boston, Boston, MA, pp. 115 133.
39. Vardi, Y. (1989 Multiplicative censoring, renewal process, deconvolution and decreasing density: nonparametric estimation, Biometrika 76, 751 761.
40. Williamson R. E. (1956 Multiply monotone functions and their Laplace transforms, Duke Mathematical Journal 23, 189 207.
41. Woodroofe, M. J. Sun (1993 A penalized maximum likelihood estimate of f(0+) when f is nonincreasing, Statistics Sinica 3, 501 515.
42. Zeidler, E. (1985 Nonlinear functional analysis and its applications. III, Springer-Verlag, New york : Variational methods and optimization Translated from German by Leo F. Boron.