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

Author

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  • Vladimir Koltchinskii

    (Georgia Institute of Technology)

  • Mayya Zhilova

    (Georgia Institute of Technology)

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.

Suggested Citation

  • Vladimir Koltchinskii & Mayya Zhilova, 2021. "Estimation of Smooth Functionals of Location Parameter in Gaussian and Poincaré Random Shift Models," Sankhya A: The Indian Journal of Statistics, Springer;Indian Statistical Institute, vol. 83(2), pages 569-596, August.
    • Handle: RePEc:spr:sankha:v:83:y:2021:i:2:d:10.1007_s13171-020-00232-1
    DOI: 10.1007/s13171-020-00232-1
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    References listed on IDEAS

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    1. Rajarshi Mukherjee & Whitney K. Newey & James Robins, 2017. "Semiparametric efficient empirical higher order influence function estimators," CeMMAP working papers CWP30/17, Centre for Microdata Methods and Practice, Institute for Fiscal Studies.
    2. Robins, James M. & Li, Lingling & Tchetgen, Eric Tchetgen & van der Vaart, Aad, 2016. "Asymptotic normality of quadratic estimators," Stochastic Processes and their Applications, Elsevier, vol. 126(12), pages 3733-3759.
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