## Abstract

Let ${\mathit{X}}_{1},\dots ,{\mathit{X}}_{\mathit{n}}$ be i.i.d. random variables sampled from a normal distribution $\mathit{N}(\mathit{\mu},\mathrm{\Sigma})$ in ${\mathbb{R}}^{\mathit{d}}$ with unknown parameter $\mathit{\theta}=(\mathit{\mu},\mathrm{\Sigma})\in \mathrm{\Theta}:={\mathbb{R}}^{\mathit{d}}\times {\mathcal{C}}_{+}^{\mathit{d}}$, where ${\mathcal{C}}_{+}^{\mathit{d}}$ is the cone of positively definite covariance operators in ${\mathbb{R}}^{\mathit{d}}$. Given a smooth functional $\mathit{f}:\mathrm{\Theta}\mapsto {\mathbb{R}}^{1}$, the goal is to estimate $\mathit{f}(\mathit{\theta})$ based on ${\mathit{X}}_{1},\dots ,{\mathit{X}}_{\mathit{n}}$. Let

$$\mathrm{\Theta}(\mathit{a};\mathit{d}):={\mathbb{R}}^{\mathit{d}}\times \{\mathrm{\Sigma}\in {\mathcal{C}}_{+}^{\mathit{d}}:\mathit{\sigma}(\mathrm{\Sigma})\subset [1/\mathit{a},\mathit{a}]\},\phantom{\rule{1em}{0ex}}\mathit{a}\ge 1,$$

where $\mathit{\sigma}(\mathrm{\Sigma})$ is the spectrum of covariance Σ. Let $\stackrel{\u02c6}{\mathit{\theta}}:=(\stackrel{\u02c6}{\mathit{\mu}},\stackrel{\u02c6}{\mathrm{\Sigma}})$, where $\stackrel{\u02c6}{\mathit{\mu}}$ is the sample mean and $\stackrel{\u02c6}{\mathrm{\Sigma}}$ is the sample covariance, based on the observations ${\mathit{X}}_{1},\dots ,{\mathit{X}}_{\mathit{n}}$. For an arbitrary functional $\mathit{f}\in {\mathit{C}}^{\mathit{s}}(\mathrm{\Theta})$, $\mathit{s}=\mathit{k}+1+\mathit{\rho},\mathit{k}\ge 0,\mathit{\rho}\in (0,1]$, we define a functional ${\mathit{f}}_{\mathit{k}}:\mathrm{\Theta}\mapsto \mathbb{R}$ such that

$$\underset{\mathit{\theta}\in \mathrm{\Theta}(\mathit{a};\mathit{d})}{sup}{\Vert {\mathit{f}}_{\mathit{k}}(\stackrel{\u02c6}{\mathit{\theta}})-\mathit{f}(\mathit{\theta})\Vert}_{{\mathit{L}}_{2}({\mathbb{P}}_{\mathit{\theta}})}{\lesssim}_{\mathit{s},\mathit{\beta}}\Vert \mathit{f}{\Vert}_{{\mathit{C}}^{\mathit{s}}(\mathrm{\Theta})}\left[\right(\frac{\mathit{a}}{\sqrt{\mathit{n}}}\vee {\mathit{a}}^{\mathit{\beta}\mathit{s}}{\left(\sqrt{\frac{\mathit{d}}{\mathit{n}}}\right)}^{\mathit{s}})\wedge 1],$$

where $\mathit{\beta}=1$ for $\mathit{k}=0$ and $\mathit{\beta}>\mathit{s}-1$ is arbitrary for $\mathit{k}\ge 1$. This error rate is minimax optimal and similar bounds hold for more general loss functions. If $\mathit{d}={\mathit{d}}_{\mathit{n}}\le {\mathit{n}}^{\mathit{\alpha}}$ for some $\mathit{\alpha}\in (0,1)$ and $\mathit{s}\ge \frac{1}{1-\mathit{\alpha}}$, the rate becomes $\mathit{O}({\mathit{n}}^{-1/2})$. Moreover, for $\mathit{s}>\frac{1}{1-\mathit{\alpha}}$, the estimator ${\mathit{f}}_{\mathit{k}}(\stackrel{\u02c6}{\mathit{\theta}})$ is shown to be asymptotically efficient. The crucial part of the construction of estimator ${\mathit{f}}_{\mathit{k}}(\stackrel{\u02c6}{\mathit{\theta}})$ is a bias reduction method studied in the paper for more general statistical models than normal.

## Funding Statement

The first author was supported in part by NSF Grant DMS-1810958.

The second author was supported in part by NSF Grant DMS-1712990.

## Acknowledgments

The authors are very thankful to the Associate Editor and anonymous referees for helpful comments and suggestions.

## Citation

Vladimir Koltchinskii. Mayya Zhilova. "Estimation of smooth functionals in normal models: Bias reduction and asymptotic efficiency." Ann. Statist. 49 (5) 2577 - 2610, October 2021. https://doi.org/10.1214/20-AOS2047

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