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.
Accepté le :
DOI : 10.1051/ps/2018027
Mots-clés : Inverse problem, unknown eigenvalues, minimax theory, rate optimality, truncated series estimator, non-parametric testing, goodness-of-fit testing, signal detection
@article{PS_2019__23__524_0,
author = {Kroll, Martin},
title = {Rate optimal estimation of quadratic functionals in inverse problems with partially unknown operator and application to testing problems},
journal = {ESAIM: Probability and Statistics},
pages = {524--551},
publisher = {EDP-Sciences},
volume = {23},
year = {2019},
doi = {10.1051/ps/2018027},
mrnumber = {3990654},
zbl = {1422.62129},
language = {en},
url = {http://archive.numdam.org/articles/10.1051/ps/2018027/}
}
TY - JOUR AU - Kroll, Martin TI - Rate optimal estimation of quadratic functionals in inverse problems with partially unknown operator and application to testing problems JO - ESAIM: Probability and Statistics PY - 2019 SP - 524 EP - 551 VL - 23 PB - EDP-Sciences UR - http://archive.numdam.org/articles/10.1051/ps/2018027/ DO - 10.1051/ps/2018027 LA - en ID - PS_2019__23__524_0 ER -
%0 Journal Article %A Kroll, Martin %T Rate optimal estimation of quadratic functionals in inverse problems with partially unknown operator and application to testing problems %J ESAIM: Probability and Statistics %D 2019 %P 524-551 %V 23 %I EDP-Sciences %U http://archive.numdam.org/articles/10.1051/ps/2018027/ %R 10.1051/ps/2018027 %G en %F PS_2019__23__524_0
Kroll, Martin. Rate optimal estimation of quadratic functionals in inverse problems with partially unknown operator and application to testing problems. ESAIM: Probability and Statistics, Tome 23 (2019), pp. 524-551. doi : 10.1051/ps/2018027. http://archive.numdam.org/articles/10.1051/ps/2018027/
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