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Let $\bx_j = \btheta +\bep_j, j=1,...,n$, be observations of an unknownparameter $\btheta$ in a Euclidean or separable Hilbert space $\scrH$, where$\bep_j$ are noises as random elements in $\scrH$ from a general distribution.We study the estimation of $f(\btheta)$ for a given functional$f:\scrH\rightarrow \RR$ based on $\bx_j$'s. The key element of our approach isa new method which we call High-Order Degenerate Statistical Expansion. Itleverages the use of classical multivariate Taylor expansion and degenerate$U$-statistic and yields an elegant explicit formula. In the univariate case of$\scrH=\R$, the formula expresses the error of the proposed estimator as a sumof order $k$ degenerate $U$-products of the noises with coefficient$f^{(k)}(\btheta)/k!$ and an explicit remainder term in the form of theRiemann-Liouville integral as in the Taylor expansion around the true$\btheta$. For general $\scrH$, the formula expresses the estimation error interms of the inner product of $f^{(k)}(\btheta)/k!$ and the average of thetensor products of $k$ noises with distinct indices and a parallel extension ofthe remainder term from the univariate case. This makes the proposed method anatural statistical version of the classical Taylor expansion. The proposedestimator can be viewed as a jackknife estimator of an ideal degenerateexpansion of $f(\cdot)$ around the true $\btheta$ with the degenerate$U$-product of the noises, and can be approximated by bootstrap. Thus, thejackknife, bootstrap and Taylor expansion approaches all converge to theproposed estimator. We develop risk bounds for the proposed estimator and acentral limit theorem under a second moment condition (even in expansions ofhigher than the second order). We apply this new method to generalize severalexisting results with smooth and nonsmooth $f$ to universal $\bep_j$'s withonly minimum moment constraints.

High-Order Statistical Functional Expansion and Its Application To Some Nonsmooth Problems