Abstract
We consider the related problems of estimating the -norm and the squared -norm in sparse linear regression with unknown variance, as well as the problem of testing the hypothesis that the regression parameter is null under sparse alternatives with separation. We establish the minimax optimal rates of estimation (respectively, testing) in these three problems.
Funding Statement
Y. Wang is partially supported by Tsinghua New Faculty Start-up Fund and the 2030 Innovation Megaprojects of China (Programme on New Generation Artificial Intelligence) Grant No. 2021AAA0150000. The work of O. Collier was supported by the French National Research Agency (ANR) under the grant Labex MME-DII (ANR-11-LBX-0023-01). The work of A.B.Tsybakov was supported by GENES and by ANR under the grant Labex Ecodec (ANR-11-LABEX-0047). The work of A. Carpentier is partially supported by the Deutsche Forschungsgemeinschaft (DFG) Emmy Noether grant MuSyAD (CA 1488/1-1), by the DFG - 314838170, GRK 2297 MathCoRe, by the FG DFG, by the DFG CRC 1294 ‘Data Assimilation’, Project A03, by the Forschungsgruppe FOR 5381 “Mathematische Statistik im Informationszeitalter – Statistische Effizienz und rechentechnische Durchführbarkeit”, Project 02, by the Agence Nationale de la Recherche (ANR) and the DFG on the French-German PRCI ANR ASCAI CA 1488/4-1 “Aktive und Batch-Segmentierung, Clustering und Seriation: Grundlagen der KI” and by the UFA-DFH through the French-German Doktorandenkolleg CDFA 01-18 and by the SFI Sachsen-Anhalt for the project RE-BCI.
Citation
Alexandra Carpentier. Olivier Collier. Laetitia Comminges. Alexandre B. Tsybakov. Yuhao Wang. "Estimation of the -norm and testing in sparse linear regression with unknown variance." Bernoulli 28 (4) 2744 - 2787, November 2022. https://doi.org/10.3150/21-BEJ1436
