A new large deviation inequality for U-statistics of order 2

Bretagnolle, Jean

doi:10.1051/ps:1999107

Cited by 10 publications

(8 citation statements)

References 4 publications

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“…3) In order to prove Proposition 2, we use an exponential inequality with explicit constants for U -statistics of order 2 due to Houdré and Reynaud [13]. It is worth mentioning the paper by Giné, Latala and Zinn [11] where an exponential inequality for general U -statistics is given, and the paper by Bretagnolle [5] where an exponential inequality for U -statistics of order 2 is also established. 4) We could derive from the explicit constants given in Houdré and Reynaud's inequality an upper bound for κ 0 , but this upper bound would be very large.…”

Section: Commentsmentioning

confidence: 99%

Adaptive estimation of a quadratic functional of a density by model selection

2005

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Abstract.We consider the problem of estimating the integral of the square of a density f from the observation of a n sample. Our method to estimate R f 2 (x)dx is based on model selection via some penalized criterion. We prove that our estimator achieves the adaptive rates established by Efroimovich and Low on classes of smooth functions. A key point of the proof is an exponential inequality for U -statistics of order 2 due to Houdré and Reynaud.Mathematics Subject Classification. 62G05, 62G20, 62J02.

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Section: Commentsmentioning

confidence: 99%

Adaptive estimation of a quadratic functional of a density by model selection

2005

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“…These types of inequalities are already present in Hoeffding seminal papers [6], [7] and have seen further development since then. For example, exponential bounds were obtained (in the (sub)Gaussian case) by Hanson and Wright [5], by Bretagnolle [1], and most recently by Giné, Lata la, and Zinn [4] (and the many references therein). As indicated in [4], the exponential bound there is optimal since it involves a mixture of exponents corresponding to a Gaussian chaos of order two behavior, and (up to logarithmic factors) to the product of a normal and of a Poisson random variable and to the product of two independent Poisson random variables.…”

Section: Introductionmentioning

confidence: 99%

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

Houdré

Reynaud-Bouret²

2003

Stochastic Inequalities and Applications

103

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“…Baraud (2010) lists some statistical tasks relying on such deviation bounds including hypothesis testing for linear models or linear model selection. Some concentration bounds for U-statistics are available in Bretagnolle (1999), Giné et al (2000), Houdré and Reynaud-Bouret (2003). Limit theorems for quadratic forms can be found e.g.…”

Section: Chaptermentioning

confidence: 99%