Volume 23, 2019
|Page(s)||524 - 551|
|Published online||09 August 2019|
Rate optimal estimation of quadratic functionals in inverse problems with partially unknown operator and application to testing problems*,**
5 avenue Henry Le Chatelier,
*** Corresponding author: email@example.com
Accepted: 4 December 2018
We consider the estimation of quadratic functionals in a Gaussian sequence model where the eigenvalues are supposed to be unknown and accessible through noisy observations only. Imposing smoothness assumptions both on the signal and the sequence of eigenvalues, we develop a minimax theory for this problem. We propose a truncated series estimator and show that it attains the optimal rate of convergence if the truncation parameter is chosen appropriately. Consequences for testing problems in inverse problems are equally discussed: in particular, the minimax rates of testing for signal detection and goodness-of-fit testing are derived.
Mathematics Subject Classification: 62G05 / 62G10
Key words: Inverse problem / unknown eigenvalues / minimax theory / rate optimality / truncated series estimator / non-parametric testing / goodness-of-fit testing / signal detection
© EDP Sciences, SMAI 2019
Current usage metrics show cumulative count of Article Views (full-text article views including HTML views, PDF and ePub downloads, according to the available data) and Abstracts Views on Vision4Press platform.
Data correspond to usage on the plateform after 2015. The current usage metrics is available 48-96 hours after online publication and is updated daily on week days.
Initial download of the metrics may take a while.