Sparse Density Estimation with Measurement Errors

Entropy

Huiming Zhang¹  Haoyu Wei²  Shouzheng Zhang³  Xiaowei Yang⁴ 

[1] Department of Mathematics, Faculty of Science and Technology, University of Macau, Macau 999078, China;Department of Statistics, North Carolina State University, Raleigh, NC 27695, USA;Graduate School of Arts and Science, Yale University, New Haven, CT 06510-8034, USA;School of Mathematics and Statistics, Chaohu University, Hefei 238000, China;

关键词: density estimation;    Elastic-net;    measurement errors;    support recovery;    multi-mode data;   

DOI  :  10.3390/e24010030

来源: DOAJ

【 摘 要 】

This paper aims to estimate an unknown density of the data with measurement errors as a linear combination of functions from a dictionary. The main novelty is the proposal and investigation of the corrected sparse density estimator (CSDE). Inspired by the penalization approach, we propose the weighted Elastic-net penalized minimal ${\ell }_2$-distance method for sparse coefficients estimation, where the adaptive weights come from sharp concentration inequalities. The first-order conditions holding a high probability obtain the optimal weighted tuning parameters. Under local coherence or minimal eigenvalue assumptions, non-asymptotic oracle inequalities are derived. These theoretical results are transposed to obtain the support recovery with a high probability. Some numerical experiments for discrete and continuous distributions confirm the significant improvement obtained by our procedure when compared with other conventional approaches. Finally, the application is performed in a meteorology dataset. It shows that our method has potency and superiority in detecting multi-mode density shapes compared with other conventional approaches.

【 授权许可】

Unknown