Uniform in bandwidth consistency of kernel-type function estimators

We introduce a general method to prove uniform in bandwidth consistency of kernel-type function estimators. Examples include the kernel density estimator, the Nadaraya-Watson regression estimator and the conditional empirical process. Our results may be useful to establish uniform consistency of data-driven bandwidth kernel-type function estimators.

[1]  Paul Deheuvels,et al.  Uniform Limit Laws for Kernel Density Estimators on Possibly Unbounded Intervals , 2000 .

[2]  Jon A. Wellner,et al.  Weak Convergence and Empirical Processes: With Applications to Statistics , 1996 .

[3]  D. Bosq,et al.  Théorie de l'estimation fonctionnelle , 1987 .

[4]  J. Staniswalis.  Local Bandwidth Selection for Kernel Estimates , 1989 .

[5]  E. Giné,et al.  Rates of strong uniform consistency for multivariate kernel density estimators , 2002 .

[6]  Jianqing Fan,et al.  Local polynomial modelling and its applications , 1994 .

[7]  Winfried Stute,et al.  The Oscillation Behavior of Empirical Processes: The Multivariate Case , 1984 .

[8]  D. Mason,et al.  Functional Laws of the Iterated Logarithm for the Increments of Empirical and Quantile Processes , 1992 .

[9]  Wolfgang Härdle,et al.  Strong uniform consistency rates for estimators of conditional functionals , 1988 .

[10]  L. Devroye,et al.  Nonparametric density estimation : the L[1] view , 1987 .

[11]  W. Stute.  A Law of the Logarithm for Kernel Density Estimators , 1982 .

[12]  M. Talagrand.  Sharper Bounds for Gaussian and Empirical Processes , 1994 .

[13]  Alan F Karr,et al.  Maximum Penalized Likelihood Estimation, Vol. I: Density Estimation , 2003 .

[14]  D. Mason,et al.  General Asymptotic Confidence Bands Based on Kernel-type Function Estimators , 2004 .

[15]  J. Marron,et al.  Uniform consistency of automatic and location-adaptive delta-sequence estimators , 1989 .

[16]  M. C. Jones,et al.  A Brief Survey of Bandwidth Selection for Density Estimation , 1996 .

[17]  H. Müller,et al.  Variable Bandwidth Kernel Estimators of Regression Curves , 1987 .

[18]  Uwe Einmahl,et al.  An Empirical Process Approach to the Uniform Consistency of Kernel-Type Function Estimators , 2000 .

[19]  V. Spokoiny,et al.  Optimal pointwise adaptive methods in nonparametric estimation , 1997 .

[20]  P. Eggermont,et al.  Maximum penalized likelihood estimation , 2001 .

[21]  D. Pollard,et al.  $U$-Processes: Rates of Convergence , 1987 .

[22]  E. Mammen,et al.  Optimal spatial adaptation to inhomogeneous smoothness: an approach based on kernel estimates with variable bandwidth selectors , 1997 .

[23]  E. Giné,et al.  On consistency of kernel density estimators for randomly censored data: rates holding uniformly over adaptive intervals , 2001 .

[24]  Winfried Stute,et al.  THE OSCILLATION BEHAVIOR OF EMPIRICAL PROCESSES , 1982 .

[25]  M. Ledoux.  On Talagrand's deviation inequalities for product measures , 1997 .

[26]  Luc Devroye,et al.  Combinatorial methods in density estimation , 2001, Springer series in statistics.

[27]  J. Lamperti.  ON CONVERGENCE OF STOCHASTIC PROCESSES , 1962 .

[28]  Peter Hall,et al.  On Global Properties of Variable Bandwidth Density Estimators , 1992 .

[29]  J. Cooper.  SINGULAR INTEGRALS AND DIFFERENTIABILITY PROPERTIES OF FUNCTIONS , 1973 .

[30]  J. Kuelbs.  Probability on Banach spaces , 1978 .

[31]  Winfried Stute,et al.  ON ALMOST SURE CONVERGENCE OF CONDITIONAL EMPIRICAL DISTRIBUTION FUNCTIONS , 1986 .

[32]  L. Breiman,et al.  Variable Kernel Estimates of Multivariate Densities , 1977 .