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Let $X_1,\dots, X_n$ be i.i.d. random variables sampled from a normaldistribution $N(\mu,\Sigma)$ in ${\mathbb R}^d$ with unknown parameter$\theta=(\mu,\Sigma)\in \Theta:={\mathbb R}^d\times {\mathcal C}_+^d,$ where${\mathcal C}_+^d$ is the cone of positively definite covariance operators in${\mathbb R}^d.$ Given a smooth functional $f:\Theta \mapsto {\mathbb R}^1,$the goal is to estimate $f(\theta)$ based on $X_1,\dots, X_n.$ Let $$\Theta(a;d):={\mathbb R}^d\times \Bigl\{\Sigma\in {\mathcal C}_+^d:\sigma(\Sigma)\subset [1/a, a]\Bigr\}, a\geq 1, $$ where $\sigma(\Sigma)$ isthe spectrum of covariance $\Sigma.$ Let $\hat \theta:=(\hat \mu, \hat\Sigma),$ where $\hat \mu$ is the sample mean and $\hat \Sigma$ is the samplecovariance, based on the observations $X_1,\dots, X_n.$ For an arbitraryfunctional $f\in C^s(\Theta),$ $s=k+1+\rho, k\geq 0, \rho\in (0,1],$ we definea functional $f_k:\Theta \mapsto {\mathbb R}$ such that \begin{align*} &\sup_{\theta\in \Theta(a;d)}\|f_k(\hat \theta)-f(\theta)\|_{L_2({\mathbbP}_{\theta})} \lesssim_{s, \beta} \|f\|_{C^{s}(\Theta)}\biggr[\biggl(\frac{a}{\sqrt{n}} \bigvee a^{\betas}\biggl(\sqrt{\frac{d}{n}}\biggr)^{s} \biggr)\wedge 1\biggr], \end{align*}where $\beta =1$ for $k=0$ and $\beta>s-1$ is arbitrary for $k\geq 1.$ Thiserror rate is minimax optimal and similar bounds hold for more general lossfunctions. If $d=d_n\leq n^{\alpha}$ for some $\alpha\in (0,1)$ and $s\geq\frac{1}{1-\alpha},$ the rate becomes $O(n^{-1/2}).$ Moreover, for$s>\frac{1}{1-\alpha},$ the estimators $f_k(\hat \theta)$ is shown to beasymptotically efficient. The crucial part of the construction of estimator$f_k(\hat \theta)$ is a bias reduction method studied in the paper for moregeneral statistical models than normal.

Estimation of Smooth Functionals in Normal Models: Bias Reduction and Asymptotic Efficiency