On local U-statistic processes and the estimation of densities of functions of several sample variables
Evarist Giné and David M. Mason
Source: Ann. Statist. Volume 35, Number 3 (2007), 1105-1145.
Abstract
A notion of local U-statistic process is introduced and central limit theorems in various norms are obtained for it. This involves the development of several inequalities for U-processes that may be useful in other contexts. This local U-statistic process is based on an estimator of the density of a function of several sample variables proposed by Frees [J. Amer. Statist. Assoc. 89 (1994) 517–525] and, as a consequence, uniform in bandwidth central limit theorems in the sup and in the Lp norms are obtained for these estimators.
Primary Subjects: 60F05, 60F15, 62E20, 62G30
Keywords: U-statistics; central limit theorems; empirical process; kernel density estimation
Full-text: Access denied (no subscription detected)
We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber.
If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text
Links and Identifiers
Permanent link to this document: http://projecteuclid.org/euclid.aos/1185304000
Digital Object Identifier: doi:10.1214/009053607000000154
References
Ahmad, I. A. and Fan, Y. (2001). Optimal bandwidths for kernel density estimators of functions of observations. Statist. Probab. Lett. 51 245--251.
Mathematical Reviews (MathSciNet):
MR1822731
Araujo, A. and Giné, E. (1980). The Central Limit Theorem for Real and Banach Valued Random Variables. Wiley, New York.
Mathematical Reviews (MathSciNet):
MR0576407
Zentralblatt MATH:
0457.60001
Arcones, M. A. (1995). A Bernstein-type inequality for $U$-statistics and $U$-processes. Statist. Probab. Lett. 22 239--247.
Mathematical Reviews (MathSciNet):
MR1323145
Bickel, P. and Ritov, Y. (1988). Estimating integrated squared density derivatives: Sharp best order of convergence estimates. Sankhyā Ser. A 50 381--393.
Mathematical Reviews (MathSciNet):
MR1065550
Deheuvels, P. and Mason, D. M. (2004). General asymptotic confidence bands based on kernel-type function estimators. Stat. Inference Stoch. Process. 7 225--277.
Mathematical Reviews (MathSciNet):
MR2111291
Digital Object Identifier: doi:10.1023/B:SISP.0000049092.55534.af
de la Peña, V. H. (1992). Decoupling and Khintchine's inequalities for $U$-statistics. Ann. Probab. 20 1877--1892.
Mathematical Reviews (MathSciNet):
MR1188046
Digital Object Identifier: doi:10.1214/aop/1176989533
Project Euclid: euclid.aop/1176989533
de la Peña, V. H. and Giné, E. (1999). Decoupling. From Dependence to Independence. Springer, New York.
Mathematical Reviews (MathSciNet):
MR1666908
Devroye, L. (1987). A Course in Density Estimation. Birkhäuser, Boston.
Mathematical Reviews (MathSciNet):
MR0891874
Zentralblatt MATH:
0617.62043
Devroye, L. (1992). A note on the usefulness of superkernels in density estimation. Ann. Statist. 20 2037--2056.
Mathematical Reviews (MathSciNet):
MR1193324
Digital Object Identifier: doi:10.1214/aos/1176348901
Project Euclid: euclid.aos/1176348901
Dudley, R. M. (1999). Uniform Central Limit Theorems. Cambridge Univ. Press.
Mathematical Reviews (MathSciNet):
MR1720712
Zentralblatt MATH:
0951.60033
Dunford, N. and Schwartz, J. T. (1988). Linear Operators. Part I. General Theory. Wiley, New York.
Mathematical Reviews (MathSciNet):
MR1009162
Zentralblatt MATH:
0635.47001
Eastwood, V. R. and Horváth, L. (1999). Limit theorems for short distances in $R^m$. Statist. Probab. Lett. 45 261--268.
Mathematical Reviews (MathSciNet):
MR1718038
Einmahl, U. and Mason, D. M. (2000). An empirical process approach to the uniform consistency of kernel-type function estimators. J. Theoret. Probab. 13 1--37.
Mathematical Reviews (MathSciNet):
MR1744994
Digital Object Identifier: doi:10.1023/A:1007769924157
Folland, G. B. (1999). Real Analysis: Modern Techniques and Their Applications, 2nd ed. Wiley, New York.
Mathematical Reviews (MathSciNet):
MR1681462
Zentralblatt MATH:
0924.28001
Frees, E. W. (1994). Estimating densities of functions of observations. J. Amer. Statist. Assoc. 89 517--525.
Mathematical Reviews (MathSciNet):
MR1294078
Digital Object Identifier: doi:10.2307/2290854
JSTOR: links.jstor.org
Giné, E. and Guillou, A. (2001). On consistency of kernel density estimators for randomly censored data: Rates holding uniformly over adaptive intervals. Ann. Inst. H. Poincaré Probab. Statist. 37 503--522.
Mathematical Reviews (MathSciNet):
MR1876841
Digital Object Identifier: doi:10.1016/S0246-0203(01)01081-0
Giné, E. and Koltchinskii, V. (2006). Concentration inequalities and asymptotic results for ratio type empirical processes. Ann. Probab. 34 1143--1216.
Mathematical Reviews (MathSciNet):
MR2243881
Digital Object Identifier: doi:10.1214/009117906000000070
Project Euclid: euclid.aop/1151418495
Giné, E., Latała, R. and Zinn, J. (2000). Exponential and moment inequalities for $U$-statistics. In High Dimensional Probability, II (E. Giné, D. M. Mason and J. A. Wellner, eds.) 13--38. Birkhäuser, Boston.
Mathematical Reviews (MathSciNet):
MR1857312
Giné, E. and Mason, D. M. (2006). Uniform in bandwidth estimation of integral functionals of the density function. Preprint.
Giné, E. and Mason, D. M. (2007). Laws of the iterated logarithm for the local $U$-statistic process. J. Theoret. Probab. To appear.
Mathematical Reviews (MathSciNet):
MR2337137
Digital Object Identifier: doi:10.1007/s10959-007-0067-0
Giné, E., Mason, D. M. and Zaitsev, A. Yu. (2003). The $L_1$-norm density estimator process. Ann. Probab. 31 719--768.
Mathematical Reviews (MathSciNet):
MR1964947
Digital Object Identifier: doi:10.1214/aop/1048516534
Project Euclid: euclid.aop/1048516534
Jain, N. (1977). Central limit theorem and related questions in Banach space. In Probability (J. L. Doob, ed.) 55--65. Amer. Math. Soc., Providence, RI.
Mathematical Reviews (MathSciNet):
MR0451328
Zentralblatt MATH:
0389.60002
Jammalamadaka, S. R. and Janson, S. (1986). Limit theorems for a triangular scheme of $U$-statistics with applications to inter-point distances. Ann. Probab. 14 1347--1358.
Mathematical Reviews (MathSciNet):
MR0866355
Digital Object Identifier: doi:10.1214/aop/1176992375
Project Euclid: euclid.aop/1176992375
Ledoux, M. and Talagrand, M. (1991). Probability in Banach Spaces. Springer, Berlin.
Mathematical Reviews (MathSciNet):
MR1102015
Zentralblatt MATH:
0748.60004
Mason, D. M. (2004). A uniform functional law of the logarithm for the local empirical process. Ann. Probab. 32 1391--1418.
Mathematical Reviews (MathSciNet):
MR2060302
Digital Object Identifier: doi:10.1214/009117904000000243
Project Euclid: euclid.aop/1084884855
Nolan, D. and Pollard, D. (1987). $U$-processes: Rates of convergence. Ann. Statist. 15 780--799.
Mathematical Reviews (MathSciNet):
MR0888439
Digital Object Identifier: doi:10.1214/aos/1176350374
Project Euclid: euclid.aos/1176350374
Parzen, E. (1962). On estimation of a probability density function and mode. Ann. Math. Statist. 33 1065--1076.
Mathematical Reviews (MathSciNet):
MR0143282
Digital Object Identifier: doi:10.1214/aoms/1177704472
Project Euclid: euclid.aoms/1177704472
Pinelis, I. F. (1995). Optimum bounds on moments of sums of independent random vectors. Siberian Adv. Math. 5 141--150.
Mathematical Reviews (MathSciNet):
MR1387858
Pisier, G. and Zinn, J. (1978). On the limit theorems for random variables with values in the spaces $L_p$, $2\leq p<\infty$. Z. Wahrsch. Verw. Gebiete 41 289--304.
Mathematical Reviews (MathSciNet):
MR0471010
Digital Object Identifier: doi:10.1007/BF00533600
Pollard, D. (1984). Convergence of Stochastic Processes. Springer, New York.
Mathematical Reviews (MathSciNet):
MR0762984
Zentralblatt MATH:
0544.60045
Rost, D. (2000). Limit theorems for smoothed empirical processes. In High Dimensional Probability, II (E. Giné, D. M. Mason and J. A. Wellner, eds.) 107--113. Birkhäuser, Boston.
Mathematical Reviews (MathSciNet):
MR1857318
Zentralblatt MATH:
0968.60023
Schick, A. and Wefelmeyer, W. (2004). Root $n$ consistent density estimators for sums of independent random variables. J. Nonparametr. Stat. 16 925--935.
Mathematical Reviews (MathSciNet):
MR2094747
Digital Object Identifier: doi:10.1080/10485250410001713990
Schick, A. and Wefelmeyer, W. (2006). Pointwise convergence rates and central limit theorems for kernel density estimators in linear processes. Statist. Probab. Lett. 76 1756--1760.
Mathematical Reviews (MathSciNet):
MR2274137
Schick, A. and Wefelmeyer, W. (2007). Root-$n$ consistent density estimators of convolutions in weighted $L_1$-norms. J. Statist. Plann. Inference 137 1765--1774.
Mathematical Reviews (MathSciNet):
MR2323861
Digital Object Identifier: doi:10.1016/j.jspi.2006.06.041
Talagrand, M. (1994). Sharper bounds for Gaussian and empirical processes. Ann. Probab. 22 28--76.
Mathematical Reviews (MathSciNet):
MR1258865
Digital Object Identifier: doi:10.1214/aop/1176988847
Project Euclid: euclid.aop/1176988847
Vakhania, N. (1981). Probability Distributions on Linear Spaces. North-Holland, Amsterdam.
Mathematical Reviews (MathSciNet):
MR0626346
Zentralblatt MATH:
0481.60002
van der Vaart, A. W. and Wellner, J. A. (1996). Weak Convergence and Empirical Processes. With Applications to Statistics. Springer, New York.
Mathematical Reviews (MathSciNet):
MR1385671
Zentralblatt MATH:
0862.60002
The Annals of Statistics
