Uniform in bandwidth estimation of integral functionals of the density function

Scandinavian Journal of Statistics

Volumn 35, Issue 4, 2008, Pages 739-761

  Giné, Evarist ^a   Mason, David M ^b  

^a UNIVERSITY OF CONNECTICUT   (United States)

^b UNIVERSITY OF DELAWARE   (United States)

Author keywords

Kernel density estimator; U statistics; Uniform in bandwidth

Indexed keywords

EID: 55849092871     PISSN: 03036898     EISSN: 14679469     Source Type: Journal    
DOI: 10.1111/j.1467-9469.2008.00600.x     Document Type: Article

References (23)

1. Bickel, P. J. Ritov, Y. (1988). Estimating integrated squared density derivatives: sharp best order of convergence estimates. Sankhyā Ser. A 50, 381 393.
2. Bickel, P. J. Ritov, Y. (2003). Nonparametric estimators that can be 'plugged in'. Ann. Statist. 31, 1033 1053.
3. Birgé, L. Massart, P. (1995). Estimation of integral functionals of a density. Ann. Probab. 23, 11 29.
4. Cheng, K. F. Serfling, R. J. (1981). On estimation of a class of efficacy-related parameters. Scand. Actuar. J. 8, 83 92.
5. de la Peña, V. Giné, E. (1999). Decoupling: from dependence to independence. Springer, New York.
6. 1 view. John Wiley and Sons, New York.
7. Dmitriev, Yu. G., Koshkin, G. M., Simakhin, V. A., Tarashenko, F. P. Shulenin, V. P. (1974). Nonparametric estimation of functionals on the basis of stationary samples [in Russian]. Izdat. Tomsk. Gos. Univ., Tomsk.
8. Du, J. Schick, A. (2007). Root-n consistency and functional central limit theorems for estimators of derivatives of convolutions of densities. Int. J. Statist. Manage. Syst. 2, 67 87.
9. Eggermont, P. P. B. LaRiccia, V. N. (1999). Best asymptotic normality of the kernel density entropy estimator for smooth densities. IEEE Trans. Inform. Theory 45, 1321 1326.
10. Giné, E. Mason, D. (2007a). On local U-statistic processes and the estimation of densities of functions of several sample variables. Ann. Statist. 35, 1105 1145.
11. Giné, E. Mason, D. (2007b). Laws of the iterated logarithm for local U-statistic processes. J. Theoret. Probab. 20, 457 485.
12. Giné, E. Nickl, R. (2008a). Uniform central limit theorems for kernel density estimators. Probab. Theory Related Fields (in press).
13. Giné, E Nickl, R. (2008b). A simple adaptive estimator of the integrated square of a density. Bernoulli 14, 47 61.
14. Grübel, R. (1994). Estimation of density functionals. Ann. Inst. Statist. Math. 46, 67 75.
15. Hall, P Marron, J. S. (1987). Estimation for integrated squared density derivatives. Statist. Probab. Lett. 6, 109 155.
16. Jurečková, J. (1969). Asymptotic linearity of a rank statistic in regression parameter. Ann. Math. Statist. 40, 1889 1900.
17. Laurent, B. (1997). Estimation of integral functionals of a density and its derivatives. Bernoulli 3, 181 211.
18. Laurent, B. (2005). Adaptive estimation of a quadratic functional of a density by model selection. ESAIM Probab. Statist. 9, 1 18 (electronic).
19. Levit, B. Ya. (1978). Asymptotically efficient estimation of nonlinear functionals [in Russian]. Problems Inform. Transmission 14, 65 72.
20. Major, P. (2006). An estimate on the supremum of a nice class of stochastic integrals and U-processes. Probab. Theory Related Fields 134, 489 537.
21. Nolan, D. Pollard, D. (1987). U-processes: rates of convergence. Ann. Statist. 15, 780 799.
22. Pollard, D. (1984). Convergence of stochastic processes. Springer, New York.
23. van der Vaart, A. Wellner, J. (1996). Weak convergence of stochastic processes with applications to statistics. Springer, New York.