{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T01:39:29Z","timestamp":1760060369485,"version":"build-2065373602"},"reference-count":19,"publisher":"MDPI AG","issue":"8","license":[{"start":{"date-parts":[[2025,8,20]],"date-time":"2025-08-20T00:00:00Z","timestamp":1755648000000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/501100013139","name":"Humanities and Social Science Fund of the Ministry of Education of China","doi-asserted-by":"publisher","award":["21YJA910001","11971291"],"award-info":[{"award-number":["21YJA910001","11971291"]}],"id":[{"id":"10.13039\/501100013139","id-type":"DOI","asserted-by":"publisher"}]},{"name":"National Natural Science Foundation of China (NSFC)","award":["21YJA910001","11971291"],"award-info":[{"award-number":["21YJA910001","11971291"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Entropy"],"abstract":"<jats:p>In many practical applications, data collected over time often exhibit autocorrelation, which, if unaccounted for, can lead to biased or misleading statistical inferences. To address this issue, we propose a varying-coefficient additive model for density-valued responses, incorporating a functional auto-regressive (FAR) error process to capture serial dependence. Our estimation procedure consists of three main steps, utilizing spline-based methods after mapping density functions into a linear space via the log-quantile density transformation. First, we obtain initial estimates of the bivariate varying-coefficient functions using a B-spline series approximation. Second, we estimate the error process from the residuals using spline smoothing techniques. Finally, we refine the estimates of the additive components by adjusting for the estimated error process. We establish theoretical properties of the proposed method, including convergence rates and asymptotic behavior. The effectiveness of our approach is further demonstrated through simulation studies and applications to real-world data.<\/jats:p>","DOI":"10.3390\/e27080882","type":"journal-article","created":{"date-parts":[[2025,8,21]],"date-time":"2025-08-21T08:40:48Z","timestamp":1755765648000},"page":"882","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Varying-Coefficient Additive Models with Density Responses and Functional Auto-Regressive Error Process"],"prefix":"10.3390","volume":"27","author":[{"given":"Zixuan","family":"Han","sequence":"first","affiliation":[{"name":"Division of Public Health Sciences, Fred Hutchinson Cancer Center, Seattle, WA 98109, USA"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Tao","family":"Li","sequence":"additional","affiliation":[{"name":"School of Statistics and Data Science, Shanghai University of Finance and Economics, Shanghai 200433, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Jinhong","family":"You","sequence":"additional","affiliation":[{"name":"School of Statistics and Data Science, Shanghai University of Finance and Economics, Shanghai 200433, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-5842-8892","authenticated-orcid":false,"given":"Narayanaswamy","family":"Balakrishnan","sequence":"additional","affiliation":[{"name":"Department of Mathematics and Statistics, McMaster University, Hamilton, ON L8S 4L8, Canada"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2025,8,20]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"1304","DOI":"10.1016\/j.ijforecast.2019.05.007","article-title":"Forecasting of density functions with an application to cross-sectional and intraday returns","volume":"35","author":"Kokoszka","year":"2019","journal-title":"Int. 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