Randomized incomplete $U$-statistics in high dimensions

This paper studies inference for the mean vector of a high-dimensional $U$-statistic. In the era of Big Data, the dimension $d$ of the $U$-statistic and the sample size $n$ of the observations tend to be both large, and the computation of the $U$-statistic is prohibitively demanding. Data-dependent inferential procedures such as the empirical bootstrap for $U$-statistics is even more computationally expensive. To overcome such computational bottleneck, incomplete $U$-statistics obtained by sampling fewer terms of the $U$-statistic are attractive alternatives. In this paper, we introduce randomized incomplete $U$-statistics with sparse weights whose computational cost can be made independent of the order of the $U$-statistic. We derive non-asymptotic Gaussian approximation error bounds for the randomized incomplete $U$-statistics in high dimensions, namely in cases where the dimension $d$ is possibly much larger than the sample size $n$, for both non-degenerate and degenerate kernels. In addition, we propose generic bootstrap methods for the incomplete $U$-statistics that are computationally much less-demanding than existing bootstrap methods, and establish finite sample validity of the proposed bootstrap methods. Our methods are illustrated on the application to nonparametric testing for the pairwise independence of a high-dimensional random vector under weaker assumptions than those appearing in the literature.