| Issue |
ESAIM: PS
Volume 23, 2019
|
|
|---|---|---|
| Page(s) | 524 - 551 | |
| DOI | https://doi.org/10.1051/ps/2018027 | |
| Published online | 09 August 2019 | |
Rate optimal estimation of quadratic functionals in inverse problems with partially unknown operator and application to testing problems*,**
ENSAE-ParisTech CREST,
5 avenue Henry Le Chatelier,
91120
Palaiseau, France.
*** Corresponding author: martin.kroll@ensae.fr
Received:
4
July
2018
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
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