Statistics Surveys

A unified treatment for non-asymptotic and asymptotic approaches to minimax signal detection

Clément Marteau and Theofanis Sapatinas

Full-text: Open access

Abstract

We are concerned with minimax signal detection. In this setting, we discuss non-asymptotic and asymptotic approaches through a unified treatment. In particular, we consider a Gaussian sequence model that contains classical models as special cases, such as, direct, well-posed inverse and ill-posed inverse problems. Working with certain ellipsoids in the space of squared-summable sequences of real numbers, with a ball of positive radius removed, we compare the construction of lower and upper bounds for the minimax separation radius (non-asymptotic approach) and the minimax separation rate (asymptotic approach) that have been proposed in the literature. Some additional contributions, bringing to light links between non-asymptotic and asymptotic approaches to minimax signal, are also presented. An example of a mildly ill-posed inverse problem is used for illustrative purposes. In particular, it is shown that tools used to derive ‘asymptotic’ results can be exploited to draw ‘non-asymptotic’ conclusions, and vice-versa.

Article information

Source
Statist. Surv., Volume 9 (2015), 253-297.

Dates
Received: January 2015
First available in Project Euclid: 19 January 2016

Permanent link to this document
https://projecteuclid.org/euclid.ssu/1453212290

Digital Object Identifier
doi:10.1214/15-SS112

Mathematical Reviews number (MathSciNet)
MR3452237

Zentralblatt MATH identifier
1329.62169

Subjects
Primary: 62G05: Estimation 62K20: Response surface designs

Keywords
Gaussian sequence models ill-posed and well-posed inverse problems minimax signal detection

Citation

Marteau, Clément; Sapatinas, Theofanis. A unified treatment for non-asymptotic and asymptotic approaches to minimax signal detection. Statist. Surv. 9 (2015), 253--297. doi:10.1214/15-SS112. https://projecteuclid.org/euclid.ssu/1453212290


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