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November 2018 Adaptive estimation of high-dimensional signal-to-noise ratios
Nicolas Verzelen, Elisabeth Gassiat
Bernoulli 24(4B): 3683-3710 (November 2018). DOI: 10.3150/17-BEJ975

Abstract

We consider the equivalent problems of estimating the residual variance, the proportion of explained variance $\eta$ and the signal strength in a high-dimensional linear regression model with Gaussian random design. Our aim is to understand the impact of not knowing the sparsity of the vector of regression coefficients and not knowing the distribution of the design on minimax estimation rates of $\eta$. Depending on the sparsity $k$ of the vector regression coefficients, optimal estimators of $\eta$ either rely on estimating the vector of regression coefficients or are based on $U$-type statistics. In the important situation where $k$ is unknown, we build an adaptive procedure whose convergence rate simultaneously achieves the minimax risk over all $k$ up to a logarithmic loss which we prove to be non avoidable. Finally, the knowledge of the design distribution is shown to play a critical role. When the distribution of the design is unknown, consistent estimation of explained variance is indeed possible in much narrower regimes than for known design distribution.

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Nicolas Verzelen. Elisabeth Gassiat. "Adaptive estimation of high-dimensional signal-to-noise ratios." Bernoulli 24 (4B) 3683 - 3710, November 2018. https://doi.org/10.3150/17-BEJ975

Information

Received: 1 March 2017; Revised: 1 July 2017; Published: November 2018
First available in Project Euclid: 18 April 2018

zbMATH: 06869889
MathSciNet: MR3788186
Digital Object Identifier: 10.3150/17-BEJ975

Rights: Copyright © 2018 Bernoulli Society for Mathematical Statistics and Probability

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Vol.24 • No. 4B • November 2018
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