Sharp upper and lower bounds on the variance of U-statistics

Abstract

For a U-statistic with a symmetric kernel of any rank, we offer new upper and lower bounds on its variance, which are uniformly sharper than those established in the classical paper by Hoeffding (1948, Theorem 5.2). Our new bounds have strong optimality properties and can tightly describe how the variance scales with the degree of the kernel. As an application, we leverage our new upper bound to improve an existing Berry-Esseen theorem for the normal approximation of incomplete U-statistics based on Bernoulli sampling, in a computationally frugal regime where the computational budget is much smaller than the available sample size.