On the risk of estimates for block decreasing densities

Journal of Multivariate Analysis

Volumn 86, Issue 1, 2003, Pages 143-165

  Biau, Gérard ^a   Devroye, Luc ^b  

^a UNIV MONTPELLIER   (France)

^b MCGILL UNIVERSITY   (Canada)

Author keywords

Bandwidth selection; Block decreasing density; Minimax risk; Multivariate density estimation; Nonparametric estimation; Variable kernel estimate

Indexed keywords

EID: 0038115271     PISSN: 0047259X     EISSN: None     Source Type: Journal    
DOI: 10.1016/S0047-259X(02)00028-3     Document Type: Article

References (35)

1. I. Abramson, On bandwidth variation in kernel estimates - a square root law, Ann. Statist. 10 (1982) 1217-1223.
2. B.C. Arnold, E. Castillo, J.M. Sarabia, Conditional Specification of Statistical Models, Springer, New York, 1999.
3. P. Assouad, Deux remarques sur l'estimation, C. R. Acad. Sci. Sér. I 10 (1982) 1217-1223.
4. L. Birgé, On estimating a density using Hellinger distance and some other strange facts, Probab. Theory Related Fields 71 (1986) 271-291.
5. L. Birgé, Estimating a density under order restrictions: nonasymptotic minimax risk, Ann. Statist. 15 (1987a) 995-1012.
6. L. Birgé, On the risk of histograms for estimating decreasing densities, Ann Statist. 15 (1987b) 1013-1022.
7. L. Birgé, The Grenander estimator: a nonasymptotic approach, Ann. Statist. 17 (1989) 1532-1549.
8. L. Breiman, W. Meisel, E. Purcell, Variable kernel estimates of multivariate densities, Technometrics 19 (1977) 135-144.
9. J. Bretagnolle, C. Huber, Estimation des densités: risque minimax, Z. Wahrscheinlichkeitstheorie Verw. Gebiete 47 (1979) 119-127.
10. T.M. Cover, Geometrical and statistical properties of systems of linear inequalities with applications in pattern recognition, IEEE Trans. Electron. Comput. 14 (1965) 326-334.
11. 1 consistency of variable kernel estimates, Ann. Statist. 13 (1985) 1041-1049.
12. L. Devroye, A Course in Density Estimation, Birkhäuser, Boston, 1987.
13. 1 View, Wiley, New York, 1985.
14. L. Devroye, A. Krzyzak, New multivariate product density estimators, 2000, unpublished.
15. L. Devroye, G. Lugosi, Variable kernel estimates: on the impossibility of tuning the parameters, in: E. Giné, D. Mason, J. Wellner (Eds.), High-Dimensional Probability II, Springer, New York, 2000, pp. 405-424.
16. L. Devroye, G. Lugosi, Combinatorial Methods in Density Estimation, Springer, New York, 2001.
17. L. Devroye, C.S. Penrod, The strong uniform convergence of multivariate variable kernel estimates, Canad. J. Statist. 14 (1986) 211-219.
18. S. Dharmadhikari, K. Joag-Dev, Unimodality, Convexity and Applications, Academic Press, New York, 1988.
19. U. Grenander, On the theory of mortality measurement, part II, Skand. Aktuarietidskrift 39 (1956) 125-153.
20. P. Groeneboom, Estimating a monotone density, in: L. Le Cam, R.A. Olshen (Eds.), Proceedings of the Berkeley Conference in Honor of Jerzy Neyman and Jack Kiefer, Vol. II, Wadsworth, Belmont, CA, 1983, pp. 539-555.
21. J.D.F. Habbema, J. Hermans, J. Remme, Variable kernel density estimation in discriminant analysis, in: L.C.A. Corstein J. Hermans (Eds.), COMPSTAT 78, Physica-Verlag, Wien, 1978.
22. P. Hall, On global properties of variable bandwidth density estimators, Ann. Statist. 20 (1992) 762-778.
23. P. Hall, J.S. Marron, Variable window width kernel estimates, Probab. Theory Related Fields 80 (1988) 37-49.
24. M. Hazelton, Bandwidth selection for local density estimation, Scand. J. Statist. 23 (1996) 221-232.
25. J.S. Marron, P. Hall, T.C. Hu, Improved variable window estimators of probability densities, Ann. Statist. 23 (1995) 1-10.
26. J. Mielniczuk, P. Sarda, P. Vieu, Local data-driven bandwidth choice for density estimation, J. Statist. Plann. Inference 23 (1989) 53-69.
27. W. Polonik, Density estimation under qualitative assumptions in higher dimensions, J. Multivariate Anal. 55 (1995a) 61-81.
28. W. Polonik, Measuring mass concentrations and estimating density contour clusters - an excess mass approach, Ann. Statist. 23 (1995b) 855-881.
29. W. Polonik, The silhouette, concentration functions and ML-density estimation under order restrictions, Ann. Statist. 26 (1998) 1857-1877.
30. S.R. Sain, D.W. Scott, On locally adaptive density estimation, J. Amer. Statist. Assoc. 91 (1996) 1525-1534.
31. S.R. Sain, D.W. Scott, Zero-bias locally adaptive density estimators, Technical Report, Rice University, Houston, 1997.
32. G.R. Terrell, D.W. Scott, Variable kernel density estimation, Ann. Statist. 20 (1992) 1236-1265.
33. E.J. Wegnan, A note on estimating a unimodal density, Ann. Math. Statist. 40 (1969) 1661-1667.
34. E.J. Wegman, Maximum likelihood estimation of a unimodal density function, Ann. Math. Statist. 41 (1970) 457-471.
35. E.J. Wegman, Maximum likelihood estimation of a ummodal density, II, Ann. Math. Statist. 41 (1970) 2160-2174.