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October 2021 Estimation of smooth functionals in normal models: Bias reduction and asymptotic efficiency
Ann. Statist. 49(5): 2577-2610 (October 2021). DOI: 10.1214/20-AOS2047

## Abstract

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

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

where $\sigma (\Sigma )$ is the spectrum of covariance Σ. 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 $f\in {C^{s}}(\Theta )$, $s=k+1+\rho ,k\ge 0,\rho \in (0,1]$, we define a functional ${f_{k}}:\Theta \mapsto \mathbb{R}$ such that

$$\underset{\theta \in \Theta (a;d)}{\sup }{\| {f_{k}}(\hat{\theta })-f(\theta )\| _{{L_{2}}({\mathbb{P}_{\theta }})}}{\lesssim _{s,\beta }}\| f{\| _{{C^{s}}(\Theta )}}[(\frac{a}{\sqrt{n}}\vee {a^{\beta s}}{(\sqrt{\frac{d}{n}})^{s}})\wedge 1],$$

where $\beta =1$ for $k=0$ and $\beta \textgreater s-1$ is arbitrary for $k\ge 1$. This error rate is minimax optimal and similar bounds hold for more general loss functions. If $d={d_{n}}\le {n^{\alpha }}$ for some $\alpha \in (0,1)$ and $s\ge \frac{1}{1-\alpha }$, the rate becomes $O({n^{-1/2}})$. Moreover, for $s\textgreater \frac{1}{1-\alpha }$, the estimator ${f_{k}}(\hat{\theta })$ 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.

## 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

## Information

Received: 1 December 2019; Revised: 1 December 2020; Published: October 2021
First available in Project Euclid: 12 November 2021

Digital Object Identifier: 10.1214/20-AOS2047

Subjects:
Primary: 62H12
Secondary: 60B20 , 62G20 , 62H25

Keywords: bias reduction , bootstrap chain , Concentration , efficiency , random homotopy , smooth functionals