Estimation of Smooth Functionals of Location Parameter in Gaussian and Poincaré Random Shift Models

Vladimir Koltchinskii () and Mayya Zhilova ()
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Vladimir Koltchinskii: Georgia Institute of Technology
Mayya Zhilova: Georgia Institute of Technology

Sankhya A: The Indian Journal of Statistics, 2021, vol. 83, issue 2, No 4, 569-596

Abstract: Abstract Let E be a separable Banach space and let f : E ↦ ℝ $f:E\mapsto {\mathbb {R}}$ be a smooth functional. We discuss a problem of estimation of f(𝜃) based on an observation X = 𝜃 + ξ, where 𝜃 ∈ E is an unknown parameter and ξ is a mean zero random noise, or based on n i.i.d. observations from the same random shift model. We develop estimators of f(𝜃) with sharp mean squared error rates depending on the degree of smoothness of f for random shift models with distribution of the noise ξ satisfying Poincaré type inequalities (in particular, for some log-concave distributions). We show that for sufficiently smooth functionals f these estimators are asymptotically normal with a parametric convergence rate. This is done both in the case of known distribution of the noise and in the case when the distribution of the noise is Gaussian with covariance being an unknown nuisance parameter.

Keywords: Smooth functionals; Efficiency; Random shift model; Poincaré inequality; Normal approximation.; Primary 62H12; Secondary 62G20, 62H25, 60B20 (search for similar items in EconPapers)
Date: 2021
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DOI: 10.1007/s13171-020-00232-1

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