Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Estimation of Smooth Functionals in Normal Models: Bias Reduction and
  Asymptotic Efficiency   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
$\theta=(\mu,\Sigma)\in \Theta:={\mathbb R}^d\times {\mathcal C}_+^d,$ 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 $\sigma(\Sigma)$ 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 $f\in C^s(\Theta),$ $s=k+1+\rho, k\geq 0, \rho\in (0,1],$ we define
a 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({\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 1\biggr], \end{align*}
where $\beta =1$ for $k=0$ and $\beta>s-1$ is arbitrary for $k\geq 1.$ This
error rate is minimax optimal and similar bounds hold for more general loss
functions. 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 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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The categories of this paper are: cmp-lg
1
The categories of this paper are: cs
5
The categories of this paper are: gr-qc
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The categories of this paper are: supr-con
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The categories of this paper are: plasm-ph
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The categories of this paper are: chem-ph
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The categories of this paper are: comp-gas
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The categories of this paper are: hep-lat
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The categories of this paper are: q-alg
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The categories of this paper are: funct-an
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The categories of this paper are: eess
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The categories of this paper are: hep-th
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The categories of this paper are: econ
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The categories of this paper are: acc-phys
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The categories of this paper are: chao-dyn
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The categories of this paper are: atom-ph