#### DMCA

## Lectures on the central limit theorem for empirical processes (1986)

Venue: | Probability and Banach Spaces |

Citations: | 135 - 9 self |

### Citations

12870 | Statistical Learning Theory.
- Vapnik
- 1998
(Show Context)
Citation Context ...orems for classical empirical processes (see e.g. Wellner [34]). For general empirical processes indexed by sets or functions some important references are [1], [2], [3], [4], [15], [18], [27], [31], =-=[33]-=-. In order to avoid measurability problems, in what follows, we will assume that the supremum over the class F or over any of the subclasses we consider is in fact a countable supremum. In this case w... |

609 |
Weak Convergence and Empirical Processes.
- Vaart, Wellner
- 1996
(Show Context)
Citation Context ...iversal constant and κ and κ ′ are constants that depend only on A and α, and we can assume κ(A, α) ≥ 1. Then, the Borell-Sudakov-Tsirel’son inequality (cf. [21], page 57, comments following (3.2) or =-=[32]-=-, Proposition A.2.1 and its proof) gives Pξ ⎧ ⎨ ⎩ |An| ≥ so that, taking t = κ 2 (A, α)δ −α/2 n with probability at least { 1 − exp Let now −κ 2 (A, α)δ −α/2 n Bn := 2 n which we decompose as Bn = n∑ ... |

422 |
Probability in Banach spaces
- Ledoux, Talagrand
- 1991
(Show Context)
Citation Context ...bstantially modifies the proof of a similar bound (simpler, but with U = ‖F ‖∞ instead of ‖F ‖ L2(P )) in[12]. In that proof, an abstract version of the square root trick (due to Ledoux and Talagrand =-=[21]-=-) was used, whereas here, as in [11], we use the Giné and Zinn [13] version of Le Cam’s square root trick. Remark. Bounds on expectations of empirical processes that take into account the norm of the ... |

420 | Decision theoretic generalizations of the PAC model for neural nets and other learning applications
- Haussler
- 1992
(Show Context)
Citation Context ...ature on ratio limit theorems for classical empirical processes (see e.g. Wellner [34]). For general empirical processes indexed by sets or functions some important references are [1], [2], [3], [4], =-=[15]-=-, [18], [27], [31], [33]. In order to avoid measurability problems, in what follows, we will assume that the supremum over the class F or over any of the subclasses we consider is in fact a countable ... |

183 |
Sharper bounds for Gaussian and empirical processes
- Talagrand
- 1994
(Show Context)
Citation Context ...iction in Statistics and in Machine Learning. The main advance on empirical process theory since 1987, when Alexander proved his results, has been Talagrand’s discovery of concentration inequalities (=-=[29]-=-,[30]). This tool allows us to handle ratios very easily by proving several simple exponential bounds expressed in terms of expectations of localized sup norms of empirical processes. These bounds are... |

180 |
New concentration inequalities in product spaces
- TALAGRAND
- 1996
(Show Context)
Citation Context ...n in Statistics and in Machine Learning. The main advance on empirical process theory since 1987, when Alexander proved his results, has been Talagrand’s discovery of concentration inequalities ([29],=-=[30]-=-). This tool allows us to handle ratios very easily by proving several simple exponential bounds expressed in terms of expectations of localized sup norms of empirical processes. These bounds are obta... |

161 | Localized rademacher complexities
- Bartlett, Bousquet, et al.
(Show Context)
Citation Context ...ory and was developed by several authors (see, e.g., Koltchinskii and Panchenko [19], Koltchinskii [20], Panchenko [25, 26] Bousquet, Koltchinskii and Panchenko [10], Bartlett, Bousquet and Mendelson =-=[5]-=- and, especially, the Ph. D. dissertations of Panchenko [24] and Bousquet [7]). A very close approach has been developed in some other statistical applications even earlier (see [22] and references th... |

156 | Empirical margin distributions and bounding the generalization error of combined classifiers
- Koltchinskii, Panchenko
- 2002
(Show Context)
Citation Context ... stratum and then collecting terms. This approach originated in the more specialized setting of statistical learning theory and was developed by several authors (see, e.g., Koltchinskii and Panchenko =-=[19]-=-, Koltchinskii [20], Panchenko [25, 26] Bousquet, Koltchinskii and Panchenko [10], Bartlett, Bousquet and Mendelson [5] and, especially, the Ph. D. dissertations of Panchenko [24] and Bousquet [7]). A... |

112 |
Some applications of concentration inequalities to statistics
- Massart
(Show Context)
Citation Context ...quet and Mendelson [5] and, especially, the Ph. D. dissertations of Panchenko [24] and Bousquet [7]). A very close approach has been developed in some other statistical applications even earlier (see =-=[22]-=- and references therein). The exponential bounds for ratios together with some new bounds on expectations of suprema of empirical processes over VC classes of functions ([29], [12], [11], [23]) allow ... |

71 |
An empirical process approach to the uniform consistency of kernel-type function estimators
- Einmahl, Mason
- 2000
(Show Context)
Citation Context ... earlier (see [22] and references therein). The exponential bounds for ratios together with some new bounds on expectations of suprema of empirical processes over VC classes of functions ([29], [12], =-=[11]-=-, [23]) allow one to obtain Alexander type theorems without any effort. The present approach may open a possibility to understand much better, and for much more general classes than VC, this important... |

45 |
Ordered linear smoothers
- Kneip
- 1994
(Show Context)
Citation Context ...cle inequality’ for a simple version of this type of problem. For a nice introduction to oracle inequalities more generally, see [16]. The flavor of our result here is somewhat akin to the results of =-=[17]-=-. For an example of some L2−type oracle inequalities see e.g. [18]. Consider the following regression model: Yi = f0(Xi)+ξi where the ξi’s are i.i.d. N(0, 1) for simplicity. For a class of functions F... |

37 |
Concentration inequalities and empirical processes theory applied to the analysis of learning algorithms,” Ph.D. dissertation, Ecole Polytechnique
- Bousquet
- 2002
(Show Context)
Citation Context ...ko [19], Koltchinskii [20], Panchenko [25, 26] Bousquet, Koltchinskii and Panchenko [10], Bartlett, Bousquet and Mendelson [5] and, especially, the Ph. D. dissertations of Panchenko [24] and Bousquet =-=[7]-=-). A very close approach has been developed in some other statistical applications even earlier (see [22] and references therein). The exponential bounds for ratios together with some new bounds on ex... |

36 | Rademacher averages and phase transitions in glivenko-cantelli classes
- Mendelson
(Show Context)
Citation Context ...er (see [22] and references therein). The exponential bounds for ratios together with some new bounds on expectations of suprema of empirical processes over VC classes of functions ([29], [12], [11], =-=[23]-=-) allow one to obtain Alexander type theorems without any effort. The present approach may open a possibility to understand much better, and for much more general classes than VC, this important class... |

31 |
Applications of Empirical Process Theory
- Geer
- 2000
(Show Context)
Citation Context ...it theorems for classical empirical processes (see e.g. Wellner [34]). For general empirical processes indexed by sets or functions some important references are [1], [2], [3], [4], [15], [18], [27], =-=[31]-=-, [33]. In order to avoid measurability problems, in what follows, we will assume that the supremum over the class F or over any of the subclasses we consider is in fact a countable supremum. In this ... |

29 |
Concentration inequalities for sub-additive functions using the entropy method
- BOUSQUET
- 2003
(Show Context)
Citation Context ... n (u2 +2ψn,q(u)) + t } 3n and E − { q,u(t) := ‖Pn − P ‖ Fq(u) ≥ E‖Pn − P ‖ Fq(u) − √ 2 t n (u2 +20ψn,q(u)) − 2t } . 3n By Talagrand’s concentration inequalities for empirical processes (see Bousquet =-=[9]-=-), we have P(E + q,u(t)) ≥ 1 − e −t and P(E − q,u(t)) ≥ 1 − e −t Let Fj := F (ρj−1,ρj] = Fq(ρj) and E + j := E + q,ρj (t +2log j), E− j := E − q,ρj (t +2log j).6 Giné, Koltchinskii, and Wellner Then ... |

28 |
Rates of growth and sample moduli for weighted empirical processes indexed by sets. Probability Theory and Related Fields
- Alexander
- 1987
(Show Context)
Citation Context ...lasses of functions In this section we combine the main results from Sections 2-4 with the moment bound in section 5 in order to obtain analogues for VC classes of functions of some of the results in =-=[2]-=- for classes of sets. In what follows the class F is assumed to be a measurable VC class of functions (as defined in Section 5) taking values between 0 and 1, and otherwise, we resume the notation set... |

25 |
2001), On consistency of kernel density estimators for randomly censored data: Rates holding uniformly over adaptive intervals, Annales de lInstitut Henri Poincaré
- Gine, Guillou
(Show Context)
Citation Context ...s even earlier (see [22] and references therein). The exponential bounds for ratios together with some new bounds on expectations of suprema of empirical processes over VC classes of functions ([29], =-=[12]-=-, [11], [23]) allow one to obtain Alexander type theorems without any effort. The present approach may open a possibility to understand much better, and for much more general classes than VC, this imp... |

20 | Some local measures of complexity of convex hulls and generalization bounds
- Bousquet, Koltchinskii, et al.
- 2002
(Show Context)
Citation Context ...zed setting of statistical learning theory and was developed by several authors (see, e.g., Koltchinskii and Panchenko [19], Koltchinskii [20], Panchenko [25, 26] Bousquet, Koltchinskii and Panchenko =-=[10]-=-, Bartlett, Bousquet and Mendelson [5] and, especially, the Ph. D. dissertations of Panchenko [24] and Bousquet [7]). A very close approach has been developed in some other statistical applications ev... |

17 |
Limit theorems for the ratio of the empirical distribution function to the true distribution function
- Wellner
- 1978
(Show Context)
Citation Context ...ems for empirical processes, and in particular, to widen the scope of their applicability. There is an extensive literature on ratio limit theorems for classical empirical processes (see e.g. Wellner =-=[34]-=-). For general empirical processes indexed by sets or functions some important references are [1], [2], [3], [4], [15], [18], [27], [31], [33]. In order to avoid measurability problems, in what follow... |

16 | Symmetrization approach to concentration inequalities in empirical processes
- Panchenko
- 2003
(Show Context)
Citation Context ... This approach originated in the more specialized setting of statistical learning theory and was developed by several authors (see, e.g., Koltchinskii and Panchenko [19], Koltchinskii [20], Panchenko =-=[25, 26]-=- Bousquet, Koltchinskii and Panchenko [10], Bartlett, Bousquet and Mendelson [5] and, especially, the Ph. D. dissertations of Panchenko [24] and Bousquet [7]). A very close approach has been developed... |

15 |
Rates of growth for weighted empirical processes
- Alexander
- 1985
(Show Context)
Citation Context ...w inequalities for ratio-type suprema of empirical processes. These general inequalities are used to prove several new limit theorems for ratio-type 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. Introduction Let F be auniformly bounded class of real valued measurable func... |

15 | Une inégalité de Bennett pour les maxima de processus empiriques - Rio - 2002 |

14 | Inequalities for uniform deviations of averages from expectations with applications to nonparametric regression
- Kohler
(Show Context)
Citation Context ...iven here, is illustrated by an example. Indeed, as an application of our general theorems we obtain an ‘oracle inequality’ in a simple but quite general non-parametric regression setting (cf., [16], =-=[18]-=-). So, the type of inequalities proved in this article may turn out to be useful for bounding errors of prediction in Statistics and in Machine Learning. The main advance on empirical process theory s... |

13 | Some extensions of an inequality of vapnik and chervonenkis
- Panchenko
- 2002
(Show Context)
Citation Context ... This approach originated in the more specialized setting of statistical learning theory and was developed by several authors (see, e.g., Koltchinskii and Panchenko [19], Koltchinskii [20], Panchenko =-=[25, 26]-=- Bousquet, Koltchinskii and Panchenko [10], Bartlett, Bousquet and Mendelson [5] and, especially, the Ph. D. dissertations of Panchenko [24] and Bousquet [7]). A very close approach has been developed... |

12 |
Oracle inequalities and nonparametric function estimation
- Johnstone
- 1998
(Show Context)
Citation Context ...lity given here, is illustrated by an example. Indeed, as an application of our general theorems we obtain an ‘oracle inequality’ in a simple but quite general non-parametric regression setting (cf., =-=[16]-=-, [18]). So, the type of inequalities proved in this article may turn out to be useful for bounding errors of prediction in Statistics and in Machine Learning. The main advance on empirical process th... |

12 | Uniform ratio limit theorems for empirical processes
- Pollard
- 1995
(Show Context)
Citation Context ...io limit theorems for classical empirical processes (see e.g. Wellner [34]). For general empirical processes indexed by sets or functions some important references are [1], [2], [3], [4], [15], [18], =-=[27]-=-, [31], [33]. In order to avoid measurability problems, in what follows, we will assume that the supremum over the class F or over any of the subclasses we consider is in fact a countable supremum. In... |

9 | The central limit theorem for weighted empirical processes indexed by sets - Alexander - 1987 |

9 | Concentration inequalities, large and moderate deviations for selfnormalized empirical processes - Bercu, Gassiat, et al. |

7 |
An inequality for uniform deviations of sample averages from their
- BARTLETT, G
- 1999
(Show Context)
Citation Context ...literature on ratio limit theorems for classical empirical processes (see e.g. Wellner [34]). For general empirical processes indexed by sets or functions some important references are [1], [2], [3], =-=[4]-=-, [15], [18], [27], [31], [33]. In order to avoid measurability problems, in what follows, we will assume that the supremum over the class F or over any of the subclasses we consider is in fact a coun... |

5 | Bounds on margin distributions in learning problems
- Koltchinskii
(Show Context)
Citation Context ...ollecting terms. This approach originated in the more specialized setting of statistical learning theory and was developed by several authors (see, e.g., Koltchinskii and Panchenko [19], Koltchinskii =-=[20]-=-, Panchenko [25, 26] Bousquet, Koltchinskii and Panchenko [10], Bartlett, Bousquet and Mendelson [5] and, especially, the Ph. D. dissertations of Panchenko [24] and Bousquet [7]). A very close approac... |

3 | A Bennett concentration inequality and its application to empirical processes. Comptes Rendus de l'Academie des Sciences - Bousquet - 2002 |

2 |
Concentration inequalities in product spaces and applications to statistical learning theory
- PANCHENKO
- 2002
(Show Context)
Citation Context ...inskii and Panchenko [19], Koltchinskii [20], Panchenko [25, 26] Bousquet, Koltchinskii and Panchenko [10], Bartlett, Bousquet and Mendelson [5] and, especially, the Ph. D. dissertations of Panchenko =-=[24]-=- and Bousquet [7]). A very close approach has been developed in some other statistical applications even earlier (see [22] and references therein). The exponential bounds for ratios together with some... |

1 | Une inégalitédeconcentration á gauche pour les processus empiriques - Klein - 2002 |

1 | Une inégalitédeBennett pour les maxima de processus empiriques Colloque en l’honneur de - Rio - 2001 |