{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,3,13]],"date-time":"2026-03-13T08:06:18Z","timestamp":1773389178384,"version":"3.50.1"},"reference-count":30,"publisher":"MDPI AG","issue":"3","license":[{"start":{"date-parts":[[2026,3,11]],"date-time":"2026-03-11T00:00:00Z","timestamp":1773187200000},"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":"crossref","award":["12301386"],"award-info":[{"award-number":["12301386"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"crossref"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Entropy"],"abstract":"<jats:p>When the number of subjects, n, is large, paired comparisons are often sparse. Here, we study statistical inference in a class of paired comparison models parameterized by a set of merit parameters, under an Erd\u00f6s\u2013R\u00e9nyi comparison graph, where the sparsity is measured by a probability pn tending to zero. We use the moment estimation base on the scores of subjects to infer the merit parameters. We establish a unified theoretical framework in which the uniform consistency and asymptotic normality of the moment estimator hold as the number of subjects goes to infinity. A key idea for the proof of the consistency is that we obtain the convergence rate of the Newton iterative sequence for solving the estimator. We use the Thurstone model to illustrate the unified theoretical results. Further extensions to a fixed sparse comparison graph are also provided. Numerical studies and real data analysis illustrate our theoretical findings.<\/jats:p>","DOI":"10.3390\/e28030314","type":"journal-article","created":{"date-parts":[[2026,3,11]],"date-time":"2026-03-11T14:43:18Z","timestamp":1773240198000},"page":"314","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Moment Estimation in Paired Comparison Models with a Growing Number of Subjects"],"prefix":"10.3390","volume":"28","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-3353-7256","authenticated-orcid":false,"given":"Qiuping","family":"Wang","sequence":"first","affiliation":[{"name":"School of Mathematics and Statistics, Zhaoqing University, Zhaoqing 526000, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Lu","family":"Pan","sequence":"additional","affiliation":[{"name":"School of Mathematics and Statistics, Shangqiu Normal University, Shangqiu 476000, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Ting","family":"Yan","sequence":"additional","affiliation":[{"name":"Department of Statistics, Central China Normal University, Wuhan 430079, China"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2026,3,11]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"2422","DOI":"10.1080\/01621459.2022.2053137","article-title":"A general pairwise comparison model for extremely sparse networks","volume":"118","author":"Han","year":"2023","journal-title":"J. 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