Let $X_1,dots, X_n$ be i.i.d. random variables sampled from a normal distribution $N(mu,Sigma)$ in ${mathbb R}^d$ with unknown parameter $theta=(mu,Sigma)in Theta:={mathbb R}^dtimes {mathcal C}_+^d,$ 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 $$ Theta(a;d):={mathbb R}^dtimes Bigl{Sigmain {mathcal C}_+^d: sigma(Sigma)subset [1/a, a]Bigr}, ageq 1, $$ where $sigma(Sigma)$ is the spectrum of covariance $Sigma.$ Let $hat theta:=(hat mu, hat Sigma),$ where $hat mu$ is the sample mean and $hat Sigma$ is the sample covariance, based on the observations $X_1,dots, X_n.$ For an arbitrary functional $fin C^s(Theta),$ $s=k+1+rho, kgeq 0, rhoin (0,1],$ we define a functional $f_k:Theta mapsto {mathbb R}$ such that begin{align*} & sup_{thetain Theta(a;d)}|f_k(hat theta)-f(theta)|_{L_2({mathbb P}_{theta})} lesssim_{s, beta} |f|_{C^{s}(Theta)} biggr[biggl(frac{a}{sqrt{n}} bigvee a^{beta s}biggl(sqrt{frac{d}{n}}biggr)^{s} biggr)wedge 1biggr], end{align*} where $beta =1$ for $k=0$ and $beta>s-1$ is arbitrary for $kgeq 1.$ This error rate is minimax optimal and similar bounds hold for more general loss functions. If $d=d_nleq n^{alpha}$ for some $alphain (0,1)$ and $sgeq frac{1}{1-alpha},$ the rate becomes $O(n^{-1/2}).$ Moreover, for $s>frac{1}{1-alpha},$ the estimators $f_k(hat theta)$ is shown to be asymptotically efficient. The crucial part of the construction of estimator $f_k(hat theta)$ is a bias reduction method studied in the paper for more general statistical models than normal.
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