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


Let X1,,Xn be i.i.d. random variables sampled from a normal distribution N(μ,Σ) in Rd with unknown parameter θ=(μ,Σ)Θ:=Rd×C+d, where C+d is the cone of positively definite covariance operators in Rd. Given a smooth functional f:ΘR1, the goal is to estimate f(θ) based on X1,,Xn. Let


where σ(Σ) is the spectrum of covariance Σ. Let θˆ:=(μˆ,Σˆ), where μˆ is the sample mean and Σˆ is the sample covariance, based on the observations X1,,Xn. For an arbitrary functional fCs(Θ), s=k+1+ρ,k0,ρ(0,1], we define a functional fk:ΘR such that


where β=1 for k=0 and β>s1 is arbitrary for k1. This error rate is minimax optimal and similar bounds hold for more general loss functions. If d=dnnα for some α(0,1) and s11α, the rate becomes O(n1/2). Moreover, for s>11α, the estimator fk(θˆ) is shown to be asymptotically efficient. The crucial part of the construction of estimator fk(θˆ) 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.


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


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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


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

MathSciNet: MR4338376
zbMATH: 1486.62163
Digital Object Identifier: 10.1214/20-AOS2047

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

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

Rights: Copyright © 2021 Institute of Mathematical Statistics


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Vol.49 • No. 5 • October 2021
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