Shrinkage Estimation of Higher Order Bochner Integrals

We consider shrinkage estimation of higher order Hilbert space valued Bochner integrals in a non-parametric setting. We propose estimators that shrink the $U$-statistic estimator of the Bochner integral towards a pre-specified target element in the Hilbert space. Depending on the degeneracy of the kernel of the $U$-statistic, we construct consistent shrinkage estimators with fast rates of convergence, and develop oracle inequalities comparing the risks of the the $U$-statistic estimator and its shrinkage version. Surprisingly, we show that the shrinkage estimator designed by assuming complete degeneracy of the kernel of the $U$-statistic is a consistent estimator even when the kernel is not complete degenerate. This work subsumes and improves upon Krikamol et al., 2016, JMLR and Zhou et al., 2019, JMVA, which only handle mean element and covariance operator estimation in a reproducing kernel Hilbert space. We also specialize our results to normal mean estimation and show that for $d\ge 3$, the proposed estimator strictly improves upon the sample mean in terms of the mean squared error.