{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,12]],"date-time":"2025-10-12T02:35:44Z","timestamp":1760236544662,"version":"build-2065373602"},"reference-count":20,"publisher":"MDPI AG","issue":"12","license":[{"start":{"date-parts":[[2021,12,10]],"date-time":"2021-12-10T00:00:00Z","timestamp":1639094400000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"the Ministry of Education Humanities and Social Sciences Research Youth Foundation","award":["19YJC910011"],"award-info":[{"award-number":["19YJC910011"]}]},{"DOI":"10.13039\/501100007129","name":"Natural Science Foundation of Shandong Province","doi-asserted-by":"publisher","award":["ZR2020MA021","ZR2018LA003"],"award-info":[{"award-number":["ZR2020MA021","ZR2018LA003"]}],"id":[{"id":"10.13039\/501100007129","id-type":"DOI","asserted-by":"publisher"}]},{"name":"the  Project of Shandong Province Higher Educational Science and Technology Program","award":["J18KB099"],"award-info":[{"award-number":["J18KB099"]}]},{"name":"Open Research Fund Program of Data Recovery Key Laboratory of Sichuan Province","award":["DRN19020"],"award-info":[{"award-number":["DRN19020"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>In this article, we focus on the efficient estimators of the derivative of the nonparametric function in the nonparametric quantile regression model. We develop two ways of combining quantile regression information to derive the estimators. One is the weighted composite quantile regression estimator based on the quantile weighted loss function; the other is the weighted quantile average estimator based on the weighted average of quantile regression estimators at a single quantile. Furthermore, by minimizing the asymptotic variance, the optimal weight vector is computed, and consequently, the optimal estimator is obtained. Furthermore, we conduct some simulations to evaluate the performance of our proposed estimators under different symmetric error distributions. Simulation studies further illustrate that both estimators work better than the local linear least square estimator for all the symmetric errors considered except the normal error, and the weighted quantile average estimator performs better than the weighted composite quantile regression estimator in most situations.<\/jats:p>","DOI":"10.3390\/sym13122387","type":"journal-article","created":{"date-parts":[[2021,12,10]],"date-time":"2021-12-10T08:17:58Z","timestamp":1639124278000},"page":"2387","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":1,"title":["Efficient Estimation for the Derivative of Nonparametric Function by Optimally Combining Quantile Information"],"prefix":"10.3390","volume":"13","author":[{"given":"Xiaoshuang","family":"Zhou","sequence":"first","affiliation":[{"name":"School of Mathematics and Big Data, Dezhou University, Dezhou 253023, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Xiulian","family":"Gao","sequence":"additional","affiliation":[{"name":"School of Mathematics and Big Data, Dezhou University, Dezhou 253023, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Yukun","family":"Zhang","sequence":"additional","affiliation":[{"name":"School of Mathematics and Big Data, Dezhou University, Dezhou 253023, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Xiuling","family":"Yin","sequence":"additional","affiliation":[{"name":"School of Mathematics and Big Data, Dezhou University, Dezhou 253023, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Yanfeng","family":"Shen","sequence":"additional","affiliation":[{"name":"School of Mathematics and Big Data, Dezhou University, Dezhou 253023, China"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2021,12,10]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"57","DOI":"10.5183\/jjscs.22.1_57","article-title":"Comparisons of B-spline procedures with kernel procedures in estimating regression functions and their derivatives","volume":"22","author":"Dou","year":"2009","journal-title":"J. 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