{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,11,27]],"date-time":"2025-11-27T20:59:18Z","timestamp":1764277158417,"version":"build-2065373602"},"reference-count":27,"publisher":"MDPI AG","issue":"8","license":[{"start":{"date-parts":[[2022,8,6]],"date-time":"2022-08-06T00:00:00Z","timestamp":1659744000000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"Natural Science Foundation of China","award":["11803159","11271136","81530086","11671303","11201345","71771089","B14019","2015BGL001","17ZDA091"],"award-info":[{"award-number":["11803159","11271136","81530086","11671303","11201345","71771089","B14019","2015BGL001","17ZDA091"]}]},{"name":"the 111 Project of China","award":["11803159","11271136","81530086","11671303","11201345","71771089","B14019","2015BGL001","17ZDA091"],"award-info":[{"award-number":["11803159","11271136","81530086","11671303","11201345","71771089","B14019","2015BGL001","17ZDA091"]}]},{"name":"Shanghai Philosophy and Social Science Foundation","award":["11803159","11271136","81530086","11671303","11201345","71771089","B14019","2015BGL001","17ZDA091"],"award-info":[{"award-number":["11803159","11271136","81530086","11671303","11201345","71771089","B14019","2015BGL001","17ZDA091"]}]},{"name":"the National Social Science Foundation Key Program of China","award":["11803159","11271136","81530086","11671303","11201345","71771089","B14019","2015BGL001","17ZDA091"],"award-info":[{"award-number":["11803159","11271136","81530086","11671303","11201345","71771089","B14019","2015BGL001","17ZDA091"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>In multiple response regression, the reduced rank regression model is an effective method to reduce the number of model parameters and it takes advantage of interrelation among the response variables. To improve the prediction performance of the multiple response regression, a method for the sparse robust reduced rank regression with covariance estimation(Cov-SR4) is proposed, which can carry out variable selection, outlier detection, and covariance estimation simultaneously. The random error term of this model follows a multivariate normal distribution which is a symmetric distribution and the covariance matrix or precision matrix must be a symmetric matrix that reduces the number of parameters. Both the element-wise penalty function and row-wise penalty function can be used to handle different types of outliers. A numerical algorithm with a covariance estimation method is proposed to solve the robust sparse reduced rank regression. We compare our method with three recent reduced rank regression methods in a simulation study and real data analysis. Our method exhibits competitive performance both in prediction error and variable selection accuracy.<\/jats:p>","DOI":"10.3390\/sym14081617","type":"journal-article","created":{"date-parts":[[2022,8,10]],"date-time":"2022-08-10T09:47:06Z","timestamp":1660124826000},"page":"1617","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":1,"title":["Robust Sparse Reduced-Rank Regression with Response Dependency"],"prefix":"10.3390","volume":"14","author":[{"given":"Wenchen","family":"Liu","sequence":"first","affiliation":[{"name":"School of Statistics and Mathematics, Interdisciplinary Research Institute of Data Science, Shanghai Lixin University of Accounting and Finance, Shanghai 201209, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Guanfu","family":"Liu","sequence":"additional","affiliation":[{"name":"School of Statistics and Information, Shanghai University of International Business and Economics, Shanghai 201620, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Yincai","family":"Tang","sequence":"additional","affiliation":[{"name":"KLATASDS-MOE, School of Statistics, East China Normal University, Shanghai 200241, China"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2022,8,6]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"65","DOI":"10.1109\/TR.2019.2948173","article-title":"A unified model for system reliability evaluation under dynamic operating conditions","volume":"70","author":"Ancha","year":"2021","journal-title":"IEEE Trans Reliab."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"108136","DOI":"10.1016\/j.ress.2021.108136","article-title":"Modelling and estimation of system reliability under dynamic operating environments and lifetime ordering constraints","volume":"218","author":"Chunling","year":"2022","journal-title":"Reliab. 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