Exponential and Moment Inequalities for U-Statistics

By Evarist Gin'e, Rafal Latala, Joel C. Zinn. 2000

228 citations recorded by the source corpus.

A Bernstein-type exponential inequality for (generalized) canonical U-statistics of order 2 is obtained, and the Rosenthal and Hoffmann-Jorgensen inequalities for sums of independent random variables are extended to (generalized) U-statistics of any order whose kernels are either nonnegative or canonical.

Subfields: Statistics and Probability

View Paper: https://www.semanticscholar.org/paper/867d1af605daffa46aab587a255eeea7ec4d62bc

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Citations in the corpus

Hoeffding's inequality for sums of weakly dependent random variables (2015, background): “[13], Hoeffding [16], Janson [20], Joly et al.”
GLOBAL UNIFORM RISK BOUNDS FOR WAVELET DECONVOLUTION ESTIMATORS (2011, unknown): “2dv =2−jkgk ∞ ≤2 −jG2≡σ2, a bound which does not depend on h. The claim for general pfollows from standard arguments for uniformly bounded empirical processes, using, for instance, Proposition 3.1 in [18]. We nowprove thesecond statement. For every u′ >0, Talagrand’s inequality in Bousquet’s vers”
Gaussian and bootstrap approximations for high-dimensional U-statistics and their applications (2016, background): “When p (and therefore d) is fixed, the classical central limit theorems (CLT) for approximating Tn by a Gaussian random vector T \ n ∼ N(0,Γ), where Γ = Cov(g(X1)) and g(X1) = E[h(X1, X2)|X1] − θ, have been extensively studied in literature [36, 3, 39, 33, 26, 37, 34, 67, 3, 75, 32, 38, 39, 67].”
On local U-statistic processes and the estimation of densities of functions of several sample variables (2007, methodology): “We note that this theorem could have been proved, with only formal changes, for Hi depending on the subindices i. Giné, Lata la and Zinn, unpublished, have a moment inequality for B-valued canonical U -statistics of order 2 without boundedness assumptions that contains (3.32) for k = 2.”
A simple adaptive estimator of the integrated square of a density (2008, unknown): “We refer, for example, to Chapter III in Malliavin (1995) or Chapter 8 in Folland (1999) .”
Isotonic regression in general dimensions (2017, unknown): “(30) By a moment inequality for empirical processes (Giné, Lata la and Zinn, 2000, Proposition 3.1) and (28), we have E {( sup f∈Fd,ℓ ∣ ∣ ∣ ∣ 1 n1/2 n ∑ i=1 ǫif(Xi) ∣ ∣ ∣ ∣ )4} .”
A tail inequality for suprema of unbounded empirical processes with applications to Markov chains (2007, methodology): “Such applications appeared in the proof of the aforementioned moment estimates for empirical processes by Giné, Lata la, Zinn [5], in the proof of Theorem 5 and recently in the proof of Fuk-Nagaev type inequalities for empirical processes used by Einmahl and Li to investigate generalized laws of th”
Moment inequalities for functions of independent random variables (2005, background): “MR615434 [16] Giné, E., Lata la, R. and Zinn, J. (2000).”
Exponential Inequalities, with Constants, for U-statistics of Order Two (2003, background): “A canonical U-statistics of order two is generally defined for all positive integer n as (2.1) n ∑ i=1 n ∑ j=1 fi,j(Ti, Tj), where the fi,j : R × R → R are Borel measurable functions.”
Ranking and empirical minimization of U-statistics (2006, result): “We also refer to the corresponding results obtained for U -statistics by Adamczak [1], Giné, Latala and Zinn [18] and Houdré and Reynaud-Bouret [24].”
Adaptive density estimation: a curse of support? ✩ (2009, unknown): “We use Theorem 3.1 of Houdr´e and Reynaud-Bouret (2003) combined with the appropriate choice of constants derived by Klein and Rio (2005): for any u > 0 and any τ > 0, Note that similar results with unknown constants have been derived in the seminal work by Gin´e et al. (2000).”
Adaptive confidence sets in $$L^2$$ (2011, background): “The following Bernstein-type inequality for canonical U -statistics of order two is due to Giné, Latala and Zinn [12], with refinements about the numerical constants in Houdré and Reynaud-Bouret [18]: let R(x, y) be a symmetric real-valued function defined on T × T , such that ER(X,x) = 0 for all x,”