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25
Lectures on the central limit theorem for empirical processes
 Probability and Banach Spaces
, 1986
"... Abstract. Concentration inequalities are used to derive some new inequalities for ratiotype suprema of empirical processes. These general inequalities are used to prove several new limit theorems for ratiotype suprema and to recover anumber of the results from [1] and [2]. As a statistical applica ..."
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Cited by 135 (9 self)
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Abstract. Concentration inequalities are used to derive some new inequalities for ratiotype suprema of empirical processes. These general inequalities are used to prove several new limit theorems for ratiotype suprema and to recover anumber of the results from [1] and [2]. As a statistical application, an oracle inequality for nonparametric regression is obtained via ratio bounds. 1.
Uniform in bandwidth consistency of kerneltype function estimators
 Ann. Stat
, 2005
"... We introduce a general method to prove uniform in bandwidth consistency of kerneltype 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 dat ..."
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Cited by 63 (6 self)
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We introduce a general method to prove uniform in bandwidth consistency of kerneltype 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 datadriven bandwidth kerneltype function estimators. 1. Introduction and statements of main results. Let X,X1,X2,... be i.i.d. Rd, d ≥ 1, valued random variables and assume that the common distribution function of these variables has a Lebesgue density function, which we shall denote by f. A kernel K will be any measurable function which
Concentration inequalities and asymptotic results for ratio type empirical processes
 ANN. PROBAB
, 2006
"... Let F be a class of measurable functions on a measurable space (S, S) with values in [0, 1] and let Pn = n −1 n ∑ δXi i=1 be the empirical measure based on an i.i.d. sample (X1,...,Xn) from a probability distribution P on (S, S). We study the behavior of suprema of the following type: sup rn<σP f ..."
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Cited by 40 (5 self)
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Let F be a class of measurable functions on a measurable space (S, S) with values in [0, 1] and let Pn = n −1 n ∑ δXi i=1 be the empirical measure based on an i.i.d. sample (X1,...,Xn) from a probability distribution P on (S, S). We study the behavior of suprema of the following type: sup rn<σP f ≤δn Pnf − Pf  φ(σPf) where σP f ≥ Var 1/2 P f and φ is a continuous, strictly increasing function with φ(0) = 0. Using Talagrand’s concentration inequality for empirical processes, we establish concentration inequalities for such suprema and use them to derive several results about their asymptotic behavior, expressing the conditions in terms of expectations of localized suprema of empirical processes. We also prove new bounds for expected values of supnorms of empirical processes in terms of the largest σP f and the L2(P) norm of the envelope of the function class, which are especially suited for estimating localized suprema. With this technique, we extend to function classes most of the known results on ratio type suprema of empirical processes, including some of Alexander’s results for VC classes of sets. We also consider applications of these results to several important problems in nonparametric statistics and in learning theory (including general excess risk bounds in empirical risk minimization and their versions for L2regression and classification and ratio type bounds for margin distributions in classification).
On local Ustatistic processes and the estimation of densities of functions of several sample variables
, 2007
"... A notion of local Ustatistic process is introduced and central limit theorems in various norms are obtained for it. This involves the development of several inequalities for Uprocesses that may be useful in other contexts. This local Ustatistic process is based on an estimator of the density of a ..."
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Cited by 19 (4 self)
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A notion of local Ustatistic process is introduced and central limit theorems in various norms are obtained for it. This involves the development of several inequalities for Uprocesses that may be useful in other contexts. This local Ustatistic 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. 1. Introduction. Let X,X1,X2,... be i.i.d. random variables taking values in R, with common density function f and consider the kernel density estimator of f defined for t ∈ R, (1.1) fn(t,hn) = (nhn) −1
Gaussian approximation of suprema of empirical processes
, 2012
"... Abstract. We develop a new direct approach to approximating suprema of general empirical processes by a sequence of suprema of Gaussian processes, without taking the route of approximating whole empirical processes in the supremum norm. We prove an abstract approximation theorem that is applicable t ..."
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Cited by 13 (10 self)
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Abstract. We develop a new direct approach to approximating suprema of general empirical processes by a sequence of suprema of Gaussian processes, without taking the route of approximating whole empirical processes in the supremum norm. We prove an abstract approximation theorem that is applicable to a wide variety of problems, primarily in statistics. In particular, the bound in the main approximation theorem is nonasymptotic and the theorem does not require uniform boundedness of the class of functions. The proof of the approximation theorem builds on a new coupling inequality for maxima of sums of random vectors, the proof of which depends on an effective use of Stein’s method for normal approximation, and some new empirical process techniques. We study applications of this approximation theorem to local empirical processes and series estimation in nonparametric regression where the classes of functions change with the sample size and are not Donskertype. Importantly, our new technique is able to prove the Gaussian approximation for the supremum type statistics under weak regularity conditions, especially concerning the bandwidth and the number of series functions, in those examples. 1.
Weighted uniform consistency of kernel density estimators
 Ann. Probab
"... Let fn denote a kernel density estimator of a continuous density f in d dimensions, bounded and positive. Let Ψ(t) be a positive continuous function such that ‖Ψf β ‖ ∞ < ∞ for some 0 < β < 1/2. Under natural smoothness conditions, necessary and sufficient conditions for the sequence nh d n ..."
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Cited by 12 (2 self)
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Let fn denote a kernel density estimator of a continuous density f in d dimensions, bounded and positive. Let Ψ(t) be a positive continuous function such that ‖Ψf β ‖ ∞ < ∞ for some 0 < β < 1/2. Under natural smoothness conditions, necessary and sufficient conditions for the sequence nh d n 2log hd ‖Ψ(t)(fn(t) − Efn(t))‖ ∞ to be stochastically n  bounded and to converge a.s. to a constant are obtained. Also, the case of larger values of β is studied where a similar sequence with a different norming converges a.s. either to 0 or to +∞, depending on convergence or divergence of a certain integral involving the tail probabilities of Ψ(X). The results apply as well to some discontinuous not strictly positive densities. 1. Introduction. Over forty years ago, Parzen (1962
Adaptive Estimation of a Distribution Function and its Density in SupNorm Loss by Wavelet and Spline Projections
, 2008
"... Given an i.i.d. sample from a distribution F on R with uniformly continuous density p0, purelydata driven estimators are constructed that efficiently estimate F in supnorm loss, and simultaneously estimate p0 at the best possible rate of convergence over Hölder balls, also in supnorm loss. The es ..."
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Cited by 11 (7 self)
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Given an i.i.d. sample from a distribution F on R with uniformly continuous density p0, purelydata driven estimators are constructed that efficiently estimate F in supnorm loss, and simultaneously estimate p0 at the best possible rate of convergence over Hölder balls, also in supnorm loss. The estimators are obtained from applying a model selection procedure close to Lepski’s method with random thresholds to projections of the empirical measure onto spaces spanned by wavelets or Bsplines. Explicit constants in the asymptotic risk of the estimator are obtained, as well as oracletype inequalities in supnorm loss. The random thresholds are based on suprema of Rademacher processes indexed by wavelet or spline projection kernels. This requires Bernsteinanalogues of the inequalities in Koltchinskii (2006) for the deviation of suprema of empirical processes from their Rademacher symmetrizations.
A uniform functional law of the logarithm for the local empirical process
 Ann. Probab
, 2004
"... We prove a uniform functional law of the logarithm for the local empirical process. To accomplish this we combine techniques from classical and abstract empirical process theory, Gaussian distributional approximation and probability on Banach spaces. The body of techniques we develop should prove us ..."
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Cited by 10 (3 self)
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We prove a uniform functional law of the logarithm for the local empirical process. To accomplish this we combine techniques from classical and abstract empirical process theory, Gaussian distributional approximation and probability on Banach spaces. The body of techniques we develop should prove useful to the study of the strong consistency of dvariate kerneltype nonparametric function estimators. 1. Introduction. Let U, U1, U2,..., be a sequence of independent Uniform [0,1] random variables. Consider for each integer n ≥ 1 the empirical distribution function based on U1,..., Un, Gn(t) = n −1 n∑
Tail behaviour of multiple random integrals and Ustatistics ∗
"... Abstract: This paper contains sharp estimates about the distribution of multiple random integrals of functions of several variables with respect to a normalized empirical measure, about the distribution of Ustatistics and multiple Wiener–Itô integrals with respect to a white noise. It also contains ..."
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Cited by 10 (0 self)
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Abstract: This paper contains sharp estimates about the distribution of multiple random integrals of functions of several variables with respect to a normalized empirical measure, about the distribution of Ustatistics and multiple Wiener–Itô integrals with respect to a white noise. It also contains good estimates about the supremum of appropriate classes of such integrals or Ustatistics. The proof of most results is omitted, I have concentrated on the explanation of their content and the picture behind them. I also tried to explain the reason for the investigation of such questions. My goal was to yield such a presentation of the results which a nonexpert also can understand, and not only on a formal level.
KERNEL DENSITY ESTIMATORS: CONVERGENCE IN DISTRIBUTION FOR WEIGHTED SUPNORMS
"... ABSTRACT. Let fn denote a kernel density estimator of a bounded continuous density f in the real line. Let Ψ(t) beapositive continuous function such that ‖Ψf β ‖∞<∞. √Under natural smoothness conditions, necessary and sufficient conditions for the sequence nhn 2 log h −1 sup ∣ t∈R Ψ(t)(fn(t)−Efn( ..."
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Cited by 7 (1 self)
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ABSTRACT. Let fn denote a kernel density estimator of a bounded continuous density f in the real line. Let Ψ(t) beapositive continuous function such that ‖Ψf β ‖∞<∞. √Under natural smoothness conditions, necessary and sufficient conditions for the sequence nhn 2 log h −1 sup ∣ t∈R Ψ(t)(fn(t)−Efn(t)) ∣ (properly centered and normalized) to conn verge in distribution to the double exponential law are obtained. The proof is based on Gaussian approximation and a (new) limit theorem for weighted supnorms of a stationary Gaussian process. This extends well known results of Bickel and Rosenblatt to the case of weighted supnorms, with the sup taken over the whole line. In addition, all other possible limit distributions of the above sequence are identified (subject to some regularity assumptions). Runninghead: Kernel density estimators This version: May, 2002