{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,6,24]],"date-time":"2026-06-24T00:03:01Z","timestamp":1782259381076,"version":"3.54.5"},"reference-count":37,"publisher":"MDPI AG","issue":"1","license":[{"start":{"date-parts":[[2021,12,24]],"date-time":"2021-12-24T00:00:00Z","timestamp":1640304000000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/501100001809","name":"the National Natural Science Foundation of China","doi-asserted-by":"publisher","award":["11701109, 11901124"],"award-info":[{"award-number":["11701109, 11901124"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]},{"name":"the University of Macau under UM Macao Talent Programme","award":["UMMTP-2020-01"],"award-info":[{"award-number":["UMMTP-2020-01"]}]},{"name":"the General Research Project of Chaohu University","award":["XLY-201906"],"award-info":[{"award-number":["XLY-201906"]}]},{"name":"Chaohu University Applied Curriculum Development Project","award":["ch19yykc21"],"award-info":[{"award-number":["ch19yykc21"]}]},{"name":"Guangxi Science Foundation","award":["2018GXNSFAA138164"],"award-info":[{"award-number":["2018GXNSFAA138164"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Entropy"],"abstract":"<jats:p>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 \u21132-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.<\/jats:p>","DOI":"10.3390\/e24010030","type":"journal-article","created":{"date-parts":[[2021,12,24]],"date-time":"2021-12-24T08:38:46Z","timestamp":1640335126000},"page":"30","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":1,"title":["Sparse Density Estimation with Measurement Errors"],"prefix":"10.3390","volume":"24","author":[{"given":"Xiaowei","family":"Yang","sequence":"first","affiliation":[{"name":"School of Mathematics and Statistics, Chaohu University, Hefei 238000, China"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Huiming","family":"Zhang","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Faculty of Science and Technology, University of Macau, Macau 999078, China"},{"name":"UMacau Zhuhai Research Institute, Zhuhai 519031, China"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Haoyu","family":"Wei","sequence":"additional","affiliation":[{"name":"Department of Statistics, North Carolina State University, Raleigh, NC 27695, USA"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Shouzheng","family":"Zhang","sequence":"additional","affiliation":[{"name":"Graduate School of Arts and Science, Yale University, New Haven, CT 06510-8034, USA"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"1968","published-online":{"date-parts":[[2021,12,24]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"355","DOI":"10.1146\/annurev-statistics-031017-100325","article-title":"Finite mixture models","volume":"6","author":"McLachlan","year":"2019","journal-title":"Ann. 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