Robust estimation of U-statistics

Joly, Émilien; Lugosi, Gábor

doi:10.48550/arxiv.1504.04580

Cited by 1 publication

(1 citation statement)

References 13 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…, n to be a Bernoulli random variable of parameter p corresponding to the potential edges in G and F (ξ i 1 , ξ i 2 , ξ i 3 ) to be the indicator that the three potential edges, corresponding to a triplet of vertices, are all present in G thus forming a triangle. U-statistics is a class of unbiased estimators, introduced by Hoeffding [16], that has attracted considerable attention; see for example the works of Arcones [2], Giné et al [13], Hoeffding [16], Janson [20], Joly et al [21], just to name a few references. Let us bring to the reader's attention the following concentration inequality on U-statistics, which is due to Hoeffding (see [18], Section 5; see also Janson [20], Section 4).…”

Section: U -Statisticsmentioning

confidence: 99%

Hoeffding’s Inequality for Sums of Dependent Random Variables

Pelekis

Ramon

2017

Mediterr. J. Math.

View full text Add to dashboard Cite

We provide a systematic approach to deal with the following problem. Let X 1 , . . . , X n be, possibly dependent, [0, 1]-valued random variables. What is a sharp upper bound on the probability that their sum is significantly larger than their mean? In the case of independent random variables, a fundamental tool for bounding such probabilities is devised by Wassily Hoeffding. In this paper we consider analogues of Hoeffding's result for sums of dependent random variables for which we have certain information on their dependency structure. We prove a result that yields concentration inequalities for several notions of weak dependence between random variables. Additionally, we obtain a new concentration inequality for sums of, possibly dependent, [0, 1]-valued random variables, X 1 , . . . , X n , that satisfy the following condition: there exist constants γ ∈ (0, 1) and δ ∈ (0, 1] such that for every subset A ⊆ {1, . . . , n} we have E i∈A X i i / ∈A (1 − X i ) ≤ γ |A| δ n−|A| , where |A| denotes the cardinality of A. Our approach applies to several sums of weakly dependent random variables such as sums of martingale difference sequences, sums of k-wise independent random variables and U -statistics. Finally, we discuss some applications to the theory of random graphs.

show abstract