Slow rates of approximation of U-statistics and V-statistics by quadratic forms of Gaussians

2024/06/18 by Kevin Han Huang, Huang, Kevin Han, Peter Orbanz +1
Computer Science · Decision Sciences · Mathematics · #Bayesian Methods and Mixture Models #FOS: Mathematics #Probability (math.PR) #Probability and Risk Models #Statistical Methods and Inference #Statistics Theory (math.ST)

paper · pdf · doi:10.48550/arxiv.2406.12437

openalex publication_date 2024/06/18 · openalex created_date 2024/06/20 · openalex updated_date 2026/07/28

Abstract

We construct examples of degree-two U- and V-statistics of n i.i.d.~heavy-tailed random vectors in ℝ^d(n), whose ν-th moments exist for ν> 2, and provide tight bounds on the error of approximating both statistics by a quadratic form of Gaussians. In the case ν=3, the error of approximation is Θ(n^-1/12). The proof adapts a result of Huang, Austern and Orbanz [12] to U- and V-statistics. The lower bound for U-statistics is a simple example of the concept of variance domination used in [12].

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