Asymptotic normality in density support estimation

Electronic Journal of Probability

Volumn 14, Issue , 2009, Pages 2617-2635

  Biau, Gérard ^a   Cadre, Benoît ^b   Mason, David M ^c   Pelletier, Bruno ^d  

^a SORBONNE UNIVERSITÉ   (France)

^b UNIVERSITÉ PARIS SACLAY   (France)

^c UNIVERSITY OF DELAWARE   (United States)

^d UNIVERSITÉ RENNES 2   (France)

Author keywords

Central limit theorem; Nonparametric statistics; Support estimation; Tubular neighborhood

Indexed keywords

EID: 77956109927     PISSN: None     EISSN: 10836489     Source Type: Journal    
DOI: 10.1214/EJP.v14-722     Document Type: Article

References (19)

1. Bafllo, A. And Cuevas, A. (2006). Image estimators based on marked bins, Statistics, Vol. 40, pp. 277-288.
2. Bafllo, A., Cuevas, A., and Justel, A. (2000). Set estimation and nonparametric detection, Canadian Journal of Statistics, Vol. 28, pp. 765-782.
3. Beirlant, J., Györfi, L., and Lugosi, G. (1994). On the asymptotic normality of the L1- and L2-errors in histogram density estimation, Canadian Journal of Statistics, Vol. 22, pp. 309-318.
4. Beirlant, J. And Mason, D.M. (1995). On the asymptotic normality of Lp-norms of empirical functionals, Mathematical Methods of Statistics, Vol. 4, pp. 1-19.
5. Biau, G., Cadre, B., and Pelletier, B. (2008). Exact rates in density support estimation, Journal of Multivariate Analysis, Vol. 99, pp. 2185-2207.
6. Bredon, G.E. (1993). Topology and Geometry, Volume 139 of Graduate Texts in Mathematics, Springer-Verlag, New York.
7. Cuevas, A., Fraiman, R., and Rodrfguez-Casal, A. (2007). Nonparametric approach to the estimation of lengths and surface areas, The Annals of Statistics, Vol. 35, pp. 1031-1051.
8. Cuevas, A. And Rodrfguez-Casal, A. (2004). On boundary estimation, Advances in Applied Probability, Vol. 36, pp. 340-354.
9. Devroye, L. And Wise, G. (1980). Detection of abnormal behavior via nonparametric estimation of the support, SIAM Journal on Applied Mathematics, Vol. 38, pp. 480-488.
10. Gallot, S., Hulin, D. And Lafontaine, J. (2004). Riemannian Geometry, 3rd Ed., Springer-Verlag, New York.
11. Giné, E., Mason, D.M., and Zaitsev, A. (2003). The L1-norm density estimator process, The Annals of Probability, Vol. 31, pp. 719-768.
12. Gray, A. (1990). Tubes, Addison-Wesley, Redwood City, California.
13. Hall, P (1984). Random, nonuniform distribution of line segments on a circle, Stochastic Processes and their Applications, Vol. 18, pp. 239-261.
14. Hall, P (1985). Three limit theorems for vacancy in multivariate coverage problems, Journal of Multivariate Analysis, Vol. 16, pp. 211-236.
15. Hall, P (1988). Introduction to the Theory of Coverage Processes, John Wiley & Sons, New York.
16. Härdle, W, Park, B.U., and Tsybakov, A.B. (1995). Estimation of non-sharp support boundaries, Journal of Multivariate Analysis, Vol. 55, pp. 205-218.
17. Mason, D.M. And Polonik, W. (2009). Asymptotic normality of plug-in level set estimates, The Annals of Applied Probability, Vol. 19, pp. 1108-1142.
18. Molchanov, I.S. (1998). A limit theorem for solutions of inequalities, Scandinavian Journal of Statistics, Vol. 25, pp. 235-242.
19. Shergin, VV (1990). The central limit theorem for finitely dependent random variables. In: Proc. 5th Vilnius Conf. Probability and Mathematical Statistics, Vol. II, Grigelionis, B., Prohorov, Y.V, Sazonov, VV, and Statulevicius, V. (Eds.), pp. 424-431.