Let ${\mathcal P}$ be a family of probability measures on a measurable space $(S,{\mathcal A}).$ Given a Banach space $E,$ a functional $f:E\mapsto {\mathbb R}$ and a mapping $\theta: {\mathcal P}\mapsto E,$ our goal is to estimate $f(\theta(P))$ based on i.i.d. observations $X_1,\dots, X_n\sim P, P\in {\mathcal P}.$ In particular, if ${\mathcal P}=\{P_{\theta}: \theta\in \Theta\}$ is an identifiable statistical model with parameter set $\Theta\subset E,$ one can consider the mapping $\theta(P)=\theta$ for $P\in {\mathcal P}, P=P_{\theta},$ resulting in a problem of estimation of $f(\theta)$ based on i.i.d. observations $X_1,\dots, X_n\sim P_{\theta}, \theta\in \Theta.$ Given a smooth functional $f$ and estimators $\hat \theta_n(X_1,\dots, X_n), n\geq 1$ of $\theta(P),$ we use these estimators, the sample split and the Taylor expansion of $f(\theta(P))$ of a proper order to construct estimators $T_f(X_1,\dots, X_n)$ of $f(\theta(P)).$ For these estimators and for a functional $f$ of smoothness $s\geq 1,$ we prove upper bounds on the $L_p$-errors of estimator $T_f(X_1,\dots, X_n)$ under certain moment assumptions on the base estimators $\hat \theta_n.$ We study the performance of estimators $T_f(X_1,\dots, X_n)$ in several concrete problems, showing their minimax optimality and asymptotic efficiency. In particular, this includes functional estimation in high-dimensional models with many low dimensional components, functional estimation in high-dimensional exponential families and estimation of functionals of covariance operators in infinite-dimensional subgaussian models.

翻译：设 ${\mathcal P}$ 为可测空间 $(S,{\mathcal A})$ 上的一族概率测度。给定一个Banach空间 $E$、泛函 $f:E\mapsto {\mathbb R}$ 以及映射 $\theta: {\mathcal P}\mapsto E$，我们的目标是基于独立同分布观测 $X_1,\dots, X_n\sim P, P\in {\mathcal P}$ 来估计 $f(\theta(P))$。特别地，若 ${\mathcal P}=\{P_{\theta}: \theta\in \Theta\}$ 是具有参数集 $\Theta\subset E$ 的可识别统计模型，则可考虑映射 $\theta(P)=\theta$，其中 $P\in {\mathcal P}, P=P_{\theta}$，从而转化为基于独立同分布观测 $X_1,\dots, X_n\sim P_{\theta}, \theta\in \Theta$ 估计 $f(\theta)$ 的问题。对于光滑泛函 $f$ 与 $\theta(P)$ 的估计量 $\hat \theta_n(X_1,\dots, X_n), n\geq 1$，我们利用这些估计量、样本分割以及 $f(\theta(P))$ 的适当阶泰勒展开，构造 $f(\theta(P))$ 的估计量 $T_f(X_1,\dots, X_n)$。针对这些估计量及光滑度 $s\geq 1$ 的泛函 $f$，我们在基础估计量 $\hat \theta_n$ 满足特定矩假设的条件下，证明了 $T_f(X_1,\dots, X_n)$ 的 $L_p$ 误差上界。我们在若干具体问题中研究了估计量 $T_f(X_1,\dots, X_n)$ 的表现，证明其极小化最优性与渐近有效性。具体案例包括：含多低维分量的高维模型中的泛函估计、高维指数族中的泛函估计，以及无限维亚高斯模型中协方差算子泛函的估计。