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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.
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 Kendall’s process
 Journal of Multivariate Analysis
, 1996
"... Let Z1,..., Zn be a random sample of size n2 from a dvariate continuous distribution function H, and let Vi, n stand for the proportion of observations Zj, j{i, such that ZjZi componentwise. The purpose of this paper is to examine the limiting behavior of the empirical distribution function Kn deri ..."
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Cited by 30 (5 self)
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Let Z1,..., Zn be a random sample of size n2 from a dvariate continuous distribution function H, and let Vi, n stand for the proportion of observations Zj, j{i, such that ZjZi componentwise. The purpose of this paper is to examine the limiting behavior of the empirical distribution function Kn derived from the (dependent) pseudoobservations Vi, n. This random quantity is a natural nonparametric estimator of K, the distribution function of the random variable V=H(Z), whose expectation is an affine transformation of the population version of Kendall’s tau in the case d=2. Since the sample version of { is related in the same way to the mean of Kn, Genest and Rivest (1993, J. Amer. Statist. Assoc.) suggested that n[Kn(t)&K(t)] be referred to as Kendall’s process. Weak regularity conditions on K and H are found under which this centered process is asymptotically Gaussian, and an explicit expression for its limiting covariance function is given. These conditions, which are fairly easy to check, are seen to apply to large classes of multivariate distributions. 1996 Academic Press, Inc. 1.
Empirical Processes Based on PseudoObservations
 In: Asymptotic Methods in Probability and Statistics
, 1998
"... . Usually, empirical distribution functions are used to estimate the theoretical distribution function of known functions `(X) of the observable random variable X. In practice, many researchers are using empirical distribution functions constructed from residuals, which are estimations of a nonobs ..."
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Cited by 16 (4 self)
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. Usually, empirical distribution functions are used to estimate the theoretical distribution function of known functions `(X) of the observable random variable X. In practice, many researchers are using empirical distribution functions constructed from residuals, which are estimations of a nonobservable error terms in linear models. This falls under a class of more general problems in which one is interested in the estimation of the distribution function of a nonobservable random variable `(Q; X) depending on an observable random variable X together with its unknown law Q. When Q is estimated by some Qn , the quantities `(Qn ; X i ) are called pseudoobservations. Some work has been done recently when the pseudoobservations are the socalled residuals of linear models. The aim of this paper is to provide some tools to study the asymptotic behavior of empirical processes constructed from general pseudoobservations. Examples of pseudoobservations will be given together with applica...
Empirical Processes Based on PseudoObservations II: the Multivariate Case
, 1998
"... this paper is to continue the study of empirical processes constructed from general pseudoobservations in the multivariate case. Examples of pseudoobservations will be given together with applications to copulas, multivariate regression and time series. 1 ..."
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Cited by 11 (2 self)
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this paper is to continue the study of empirical processes constructed from general pseudoobservations in the multivariate case. Examples of pseudoobservations will be given together with applications to copulas, multivariate regression and time series. 1
Lower Bounds for Passive and Active Learning
"... We develop unified informationtheoretic machinery for deriving lower bounds for passive and active learning schemes. Our bounds involve the socalled Alexander’s capacity function. The supremum of this function has been recently rediscovered by Hanneke in the context of active learning under the na ..."
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We develop unified informationtheoretic machinery for deriving lower bounds for passive and active learning schemes. Our bounds involve the socalled Alexander’s capacity function. The supremum of this function has been recently rediscovered by Hanneke in the context of active learning under the name of “disagreement coefficient. ” For passive learning, our lower bounds match the upper bounds of Giné and Koltchinskii up to constants and generalize analogous results of Massart and Nédélec. For active learning, we provide first known lower bounds based on the capacity function rather than the disagreement coefficient. 1
Subsampling methods for persistent homology
, 2014
"... Persistent homology is a multiscale method for analyzing the shape of sets and functions from point cloud data arising from an unknown distribution supported on those sets. When the size of the sample is large, direct computation of the persistent homology is prohibitive due to the combinatorial nat ..."
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Cited by 1 (1 self)
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Persistent homology is a multiscale method for analyzing the shape of sets and functions from point cloud data arising from an unknown distribution supported on those sets. When the size of the sample is large, direct computation of the persistent homology is prohibitive due to the combinatorial nature of the existing algorithms. We propose to compute the persistent homology of several subsamples of the data and then combine the resulting estimates. We study the risk of two estimators and we prove that the subsampling approach carries stable topological information while achieving a great reduction in computational complexity. 1
United Arab Emirates kghoudiQuaeu.ac.ae
"... This paper is dedicated Miklos Csorgd on his 70th birthday Abstract. One often needs to estimate the distribution functions of a random vector E = H(X), where the H is unknown and might depend on the law of X. When H is estimated by some H,, using a sample XI,..., X,, the H,,(X,)'s are termed ..."
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This paper is dedicated Miklos Csorgd on his 70th birthday Abstract. One often needs to estimate the distribution functions of a random vector E = H(X), where the H is unknown and might depend on the law of X. When H is estimated by some H,, using a sample XI,..., X,, the H,,(X,)'s are termed pseudeobservations. In a semiparametric context, one often wants to estimate parameters related to the law of the nonobservable E. The transformed data Hn(X1),..., Hn(Xn) are then naturally used, introducing dependence. Classical techniques do not apply and hard work is needed to get the asymptotic behaviour of estimators and empirical processes. The aim of this paper is to give a unified treatment of inference procedures based on pseudeobservations in the multivariate setting. Exaniples of applications are given. 1