The L1-norm density estimator process

Annals of Probability

Volumn 31, Issue 2, 2003, Pages 719-768

  Giné, Evarist ^a,b   Mason, David M ^c   Zaitsev, Andrei Yu ^d  

^a UNIVERSITY OF CONNECTICUT   (United States)

^b Math Science Building 402 ^*  (United States)

^c UNIVERSITY OF DELAWARE   (United States)

^d STEKLOV MATHEMATICAL INSTITUTE   (Russian Federation)

Author keywords

Central limit theorem; Entropy; Kernel density function estimator; L1 norm; Poissonization; Weak convergence

Indexed keywords

EID: 0037998256     PISSN: 00911798     EISSN: None     Source Type: Journal    
DOI: 10.1214/aop/1048516534     Document Type: Article

References (35)

1. BARTLETT, M. S. (1938). The characteristic function of a conditional statistic. J. London Math. Soc. 13 62-67.
2. 1-norms of empirical functionals. Math. Methods Statist. 4 1-19.
3. 2-errors in histogram density estimation. Canad. J. Statist. 22 309-318.
4. BERLINET, A. and DEVROYE, L. (1994). A comparison of kernel density estimates. Publ. Inst. Statist. Univ. Paris 38 3-59.
5. BICKEL, P. J. and ROSENBLATT, M. (1973). On some global measures of the deviations of density function estimates. Ann. Statist. 1 1071-1095.
6. BILLINGSLEY, P. (1968). Convergence of Probability Measures. Wiley, New York.
7. BORISOV, I. (2002). Moment inequalities connected with accompanying Poisson laws in Abelian groups. Preprint.
8. p-norms of density estimators. Z. Wahrsch. Verw. Gebiete 80 269-291.
9. DE ACOSTA, A. (1981). Inequalities for B-valued random vectors with applications to the strong law of large numbers. Ann. Probab. 9 157-161.
10. DEHEUVELS, P. and MASON, D. M. (1992). Functional laws of the iterated logarithm for the increments of empirical and quantile processes. Ann. Probab. 20 1248-1287.
11. DE LA PEÑA, V. and GINÉ, E. (1999). Decoupling, from Dependence to Independence. Springer, New York.
12. DEVROYE, L. (1991). Exponential inequalities in nonparametric estimation. In Nonparametric Functional Estimation and Related Topics (G. Roussas, ed.) 31-44. Kluwer, Dordrecht.
13. 1 View. Wiley, New York.
14. DUDLEY, R. M. (1984). A course on empirical processes. École d'été de Probabilités de Saint-Flour XII. Lecture Notes in Math. 1097 1-142. Springer, Berlin.
15. DUDLEY, R. M. (1989). Real Analysis and Probability. Wadsworth and Brooks/Cole, Pacific Grove, CA.
16. DUNFORD, N. and SCHWARTZ, J. T. (1958). Linear Operators, Part I. Wiley, New York.
17. EGGERMONT, P. P. B. and LARICCIA, V. N. (2001). Maximum Penalized Likelihood Estimation 1. Density Estimation. Springer, New York.
18. EINMAHL, J. H. J. (1987). Multivariate Empirical Processes. CWI, Amsterdam.
19. EINMAHL, U. and MASON, D. M. (1997). Gaussian approximation of local empirical processes indexed by functions. Probab. Theory Related Fields 107 283-301.
20. FELLER, W. (1966). An Introduction to Probability Theory and Its Applications 2. Wiley, New York.
21. FOLLAND, G. B. (1999). Real Analysis, 2nd ed. Wiley, New York.
22. GINÉ, E. and ZINN, J. (1984). Some limit theorems for empirical processes. Ann. Probab. 12 929-989.
23. HOLST, L. (1979). Asymptotic normality of sum spacings. Ann. Probab. 7 1066-1072.
24. p-norms of multivariate density estimators. Ann. Statist. 19 1933-1949.
25. JOHNSON, W. B., SCHECHTMAN, G. and ZINN, J. (1985). Best constants in moment inequalities for linear combinations of independent and exchangeable random variables. Ann. Probab. 13 234-253.
26. LEDOUX, M. and TALAGRAND, M. (1991). Probability in Banach Spaces. Springer, Berlin.
27. MASON, D. M. and VAN ZWET, W. (1987). A refinement of the KMT inequality for the uniform empirical process. Ann. Probab. 15 871-884.
28. MONTGOMERY-SMITH, S. (1993). Comparison of sums of independent identically distributed random variables. Probab. Math. Statist. 14 281-285.
29. NABEYA, S. (1951). Absolute moments in 2-dimensional normal distributions. Ann. Inst. Statist. Math. 3 2-6.
30. PINELIS, I. F. (1990). Inequalities for sums of independent random vectors and their application to estimating a density. Theory Probab. Appl. 35 605-607 (translated from Russian).
31. PINELIS, I. F. (1994). On a majorization inequality for sums of independent random variables. Probab. Statist. Lett. 19 97-99.
32. PYKE, R. and SHORACK, G. R. (1968). Weak convergence of a two-sample empirical process and a new approach to Chernoff-Savage theorems. Ann. Math. Statist. 39 755-771.
33. SHERGIN, V. V. (1979). On the convergence rate in the central limit theorem for m-dependent random variables. Theory Probab. Appl. 24 782-796 (translated from Russian).
34. SWEETING, T. J. (1977). Speeds of convergence in the multidimensional central limit theorem. Ann. Probab. 5 28-41.
35. VAN DER VAART, A. and WELLNER, J. (1996). Weak Convergence and Empirical Processes. Springer, New York.