{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T01:04:05Z","timestamp":1760058245795,"version":"build-2065373602"},"reference-count":22,"publisher":"MDPI AG","issue":"3","license":[{"start":{"date-parts":[[2025,3,19]],"date-time":"2025-03-19T00:00:00Z","timestamp":1742342400000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China","doi-asserted-by":"publisher","award":["12201351","ZR2022QA013","24KJB110024","HAB202357"],"award-info":[{"award-number":["12201351","ZR2022QA013","24KJB110024","HAB202357"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100007129","name":"Natural Science Foundation of Shandong Province","doi-asserted-by":"publisher","award":["12201351","ZR2022QA013","24KJB110024","HAB202357"],"award-info":[{"award-number":["12201351","ZR2022QA013","24KJB110024","HAB202357"]}],"id":[{"id":"10.13039\/501100007129","id-type":"DOI","asserted-by":"publisher"}]},{"name":"Natural Science Foundation of the Jiangsu Higher Education Institutions of China","award":["12201351","ZR2022QA013","24KJB110024","HAB202357"],"award-info":[{"award-number":["12201351","ZR2022QA013","24KJB110024","HAB202357"]}]},{"name":"Qinglan Project of Jiangsu Province of China","award":["12201351","ZR2022QA013","24KJB110024","HAB202357"],"award-info":[{"award-number":["12201351","ZR2022QA013","24KJB110024","HAB202357"]}]},{"name":"Huai\u2019an City Science and Technology Project","award":["12201351","ZR2022QA013","24KJB110024","HAB202357"],"award-info":[{"award-number":["12201351","ZR2022QA013","24KJB110024","HAB202357"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Axioms"],"abstract":"<jats:p>A model specification test is a statistical procedure used to assess whether a given statistical model accurately represents the underlying data-generating process. The smoothing-based nonparametric specification test is widely used due to its efficiency against \u201csingular\u201d local alternatives. However, large modern datasets create various computational problems when implementing the nonparametric specification test. The divide-and-conquer algorithm is highly effective for handling large datasets, as it can break down a large dataset into more manageable datasets. By applying divide-and-conquer, the nonparametric specification test can handle the computational problems induced by the massive size of the modern datasets, leading to improved scalability and efficiency and reduced processing time. However, the selection of smoothing parameters for optimal power of the distributed algorithm is an important problem. The rate of the smoothing parameter that ensures rate optimality of the test in the context of testing the specification of a nonlinear parametric regression function is studied in the literature. In this paper, we verified the uniqueness of the rate of the smoothing parameter that ensures the rate optimality of divide-and-conquer-based tests. By employing a penalty method to select the smoothing parameter, we obtain a test with an asymptotic normal null distribution and adaptiveness properties. The performance of this test is further illustrated through numerical simulations.<\/jats:p>","DOI":"10.3390\/axioms14030228","type":"journal-article","created":{"date-parts":[[2025,3,20]],"date-time":"2025-03-20T05:03:59Z","timestamp":1742447039000},"page":"228","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Optimal Minimax Rate of Smoothing Parameter in Distributed Nonparametric Specification Test"],"prefix":"10.3390","volume":"14","author":[{"given":"Peili","family":"Liu","sequence":"first","affiliation":[{"name":"Department of Biostatistics, School of Public Health, Shandong University, Jinan 250021, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-3235-3076","authenticated-orcid":false,"given":"Yanyan","family":"Zhao","sequence":"additional","affiliation":[{"name":"Department of Biostatistics, School of Public Health, Shandong University, Jinan 250021, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Libai","family":"Xu","sequence":"additional","affiliation":[{"name":"School of Mathematical Sciences, Soochow University, Suzhou 215006, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Tao","family":"Wang","sequence":"additional","affiliation":[{"name":"School of Mathematics and Statistics, Huaiyin Normal University, Huai\u2019an 223300, China"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2025,3,19]]},"reference":[{"key":"ref_1","first-page":"755","article-title":"B-test: A Non-parametric, Low Variance Kernel Two-sample Test","volume":"26","author":"Zaremba","year":"2013","journal-title":"Adv. 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