Shrinkage Estimation of Higher Order Bochner Integrals. (arXiv:2207.06357v1 [math.ST])

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 …

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