Uniform-in-bandwidth nearest-neighbor density estimation

Statistics and Probability Letters

Volumn 83, Issue 8, 2013, Pages 1835-1843

  Ouadah, Sarah ^a  

^a SORBONNE UNIVERSITÉ   (France)

Author keywords

Convergence in probability; Functional limit laws; Nearest neighbor density estimator; Nonparametric density estimation; Uniform empirical quantile process

Indexed keywords

EID: 84877883175     PISSN: 01677152     EISSN: None     Source Type: Journal    
DOI: 10.1016/j.spl.2013.04.014     Document Type: Article

References (23)

1. k n-NN nonparametric regression and decision. Problems Control Inform. Theory/Problemy Upravlen. Teor. Inform. 1979, 8:303-311.
2. Berlinet A., Devroye L.A. Comparison of kernel density estimates. Publ. Inst. Statist. Univ. Paris 1994, 38:3-59.
3. Burba F., Ferraty F., Vieu P. k-nearest neighbour method in functional nonparametric regression. J. Nonparametr. Stat. 2009, 21:453-469.
4. Collomb G. Estimation de la régression par la méthode des k points les plus proches avec noyau: quelques propriétés de convergence ponctuelle. Lecture Notes in Mathematics 1980, vol. 821:159-175. Springer-Verlag.
5. Csörgo, M., Révész, P., 1980. A preliminary version of the preliminary version of an invariance principle for N. N. empirical density functions. Carleton Mathematical Lecture Note 27, Carleton Univ., Ottawa, pp. 91-113.
6. Deheuvels P. One bootstrap suffices to generate sharp uniform bounds in functional estimation. Kybernetica 2011, 47:881-891.
7. Deheuvels P., Derzko G. Asymptotic certainty bands for kernel density estimators based upon a bootstrap resampling scheme. Stat. Ind. Technol. 2008, 171-186. Birkhäuser Boston, Boston, MA.
8. Deheuvels P., Devroye L. Strong laws for the maximak k-spacing when k ≤ c log n. Z. Wahrscheinlichkeitstheor. Verwandte Geb. 1984, 66:315-334.
9. Deheuvels P., Mason D.M. Functional laws of the iterated logarithm for the increments of empirical and quantile processes. Ann. Probab. 1992, 20:1248-1287.
10. Deheuvels P., Mason D.M. General asymptotic confidence bands based on kernel-type function estimators. Stat. Inference Stoch. Process. 2004, 7:225-277.
11. Deheuvels P., Ouadah S. Uniform in bandwidth functionnal limit laws. J. Theor. Probab. 2011, 10.1007/s10959-011-0376-1.
12. del Barrio E., Deheuvels P., van de Geer S. Lectures on Empirical Processes. Theory and Statistical Applications. EMS Series of Lectures in Mathematics 2007.
13. Devroye L. Necessary and sufficient conditions for the pointwise convergence of nearest neighbor regression function estimates. Z. Wahrscheinlichkeitstheor. Verwandte Geb. 1982, 61:467-481.
14. Devroye L.P., Wagner T.J. The strong uniform consistency of nearest neighbor density estimates. Ann. Statist. 1977, 5:536-540.
15. Einmahl U., Mason D.M. Uniform in bandwidth consistency of kernel-type function estimators. Ann. Statist. 2005, 33:1380-1403.
16. Fix E., Hodges J.L. Discriminatory analysis. Nonparametric discrimination: consistency properties. Technical Report 4, Project No. 21-49-004 1951, USAF School of Aviation Medicine, Randoff Fiels, Texas.
17. Loftsgaarden D.O., Quesenberry C.P. A nonparametric estimate of a multivariate density function. Ann. Math. Statist. 1965, 36:1049-1051.
18. Mack Y.P. Rate of strong uniform convergence of k-NN density estimates. J. Statist. Plann. Inference 1983, 8:185-192.
19. Moore D.S., Henrichon E.G. Uniform consistency of some estimates of a density function. Ann. Math. Statist. 1969, 40:1499-1502.
20. Ralescu S.S. The law of the iterated logarithm for the multivariate nearest neighbor density estimators. J. Multivariate Anal. 1995, 53(1):159-179.
21. Shorack G.R. Kiefer's theorem via the Hungarian construction. Z. Wahrscheinlichkeitstheor. Verwandte Geb. 1982, 61:369-373.
22. Strassen V. An invariance principle for the law of the iterated logarithm. Z. Wahrscheinlichkeitstheor. Verwandte Geb. 1964, 3:211-226.
23. Viallon V. Functional limit laws for the increments of the quantile process; with applications. Electron. J. Stat. 2007, 1:496-518.