Exponential Inequalities, with Constants, for U-statistics of Order Two

By C. Houdr'e, Patricia Reynaud-Bouret. 2003

121 citations recorded by the source corpus.

Subfields: Applied Mathematics, Finance, Statistics and Probability

View Paper: https://www.semanticscholar.org/paper/3fdc5812000196dcb54561ec2478c244ea1c91a5

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Citations in the corpus

Self-concordant analysis for logistic regression (2009, methodology): “We ﬁrst apply in Section 4 the extension technique to regularization by the ℓ 2 -norm, where we consider two settings, a situation with no assumptions regarding the conditional distribution of the observations, and another one where the model is assumed well-speciﬁed and we derive asymptotic…”
Concentration inequalities for polynomials in α-sub-exponential random variables (2019, background): “and t = n, the functional Zt from [16] reads Zn = ∑ i<j aij(Xi − λi)(Xj − λj), i.”
Moment inequalities for U-statistics (2005, background): “f bounded deterministic functions with respect to stochastic processes with independent increments and uniformly bounded jumps, in the spirit of inequalities obtained by Houdr´e and Reynaud-Bouret in [7]. 2. Estimates involving suprema of empirical processes. 2.1. Basic deﬁnitions and notation. Let ”
Adaptive confidence sets in $$L^2$$ (2011, background): “The following Bernstein-type inequality for canonical U -statistics of order two is due to Giné et al. [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 E R(X, x) = 0 for all x , and le”
Adaptive density estimation: a curse of support? ✩ (2009, methodology): “Still using Theorem 3.1 of Houdré and Reynaud-Bouret (2003), we have: B2 = sup t n−1∑ l=1 E((ψ jk(t) − β jk)2(ψ jk(Xl) − β jk)2) ≤ 4(n− 1)‖ψ jk‖2∞σ2jk ≤ 4(n− 1)‖ψ jk‖4∞ Finally F = E sup i,t ∣∣∣∣∣∣ i−1∑ l=1 (ψ jk(t) − β jk)(ψ jk(Xl) − β jk) ∣∣∣∣∣∣ ≤ 2‖ψ jk‖∞E sup i ∣∣∣∣∣∣ i−1∑ l=1 (ψ jk(Xl) − β…”
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].”
A simple adaptive estimator of the integrated square of a density (2008, methodology): “We will apply Bernstein’s inequality (e.g., de la Peña and Giné (1999), page 166) to the linear part (the first probability) and its generalization for canonical U-statistics of order 2 (Giné, Latała and Zinn (2000)) with constants (Houdré and Reynaud-Bouret (2003)) to the second.”
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].”