Open Access
November 2020 Estimating linear and quadratic forms via indirect observations
Anatoli Juditsky, Arkadi Nemirovski
Bernoulli 26(4): 2639-2669 (November 2020). DOI: 10.3150/20-BEJ1200


In this paper, we further develop the approach, originating in Juditsky and Nemirovski (Ann. Statist. 37 (2009) 2278–2300), to “computation-friendly” statistical estimation via Convex Programming.Our focus is on estimating a linear or quadratic form of an unknown “signal,” known to belong to a given convex compact set, via noisy indirect observations of the signal. Classical theoretical results on the subject deal with precisely stated statistical models and aim at designing statistical inferences and quantifying their performance in a closed analytic form. In contrast to this traditional (highly instructive) descriptive framework, the approach we promote here can be qualified as operational – the estimation routines and their risks are not available “in a closed form,” but are yielded by an efficient computation. All we know in advance is that under favorable circumstances the risk of the resulting estimate, whether high or low, is provably near-optimal under the circumstances. As a compensation for the lack of “explanatory power,” this approach is applicable to a much wider family of observation schemes than those where “closed form descriptive analysis” is possible.

We discuss applications of this approach to classical problems of estimating linear forms of parameters of sub-Gaussian distribution and quadratic forms of parameters of Gaussian and discrete distributions. The performance of the constructed estimates is illustrated by computation experiments in which we compare the risks of the constructed estimates with (numerical) lower bounds for corresponding minimax risks for randomly sampled estimation problems.


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Anatoli Juditsky. Arkadi Nemirovski. "Estimating linear and quadratic forms via indirect observations." Bernoulli 26 (4) 2639 - 2669, November 2020.


Received: 1 April 2018; Revised: 1 December 2019; Published: November 2020
First available in Project Euclid: 27 August 2020

zbMATH: 07256155
MathSciNet: MR4140524
Digital Object Identifier: 10.3150/20-BEJ1200

Keywords: linear and quadratic functional estimation , linear estimation , statistical linear inverse problems

Rights: Copyright © 2020 Bernoulli Society for Mathematical Statistics and Probability

Vol.26 • No. 4 • November 2020
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