{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,12,20]],"date-time":"2025-12-20T22:08:35Z","timestamp":1766268515017,"version":"build-2065373602"},"reference-count":53,"publisher":"MDPI AG","issue":"12","license":[{"start":{"date-parts":[[2023,11,23]],"date-time":"2023-11-23T00:00:00Z","timestamp":1700697600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/501100004242","name":"Princess Nourah bint Abdulrahman University","doi-asserted-by":"publisher","award":["PNURSP2023R358","R.G.P. 1\/177\/44"],"award-info":[{"award-number":["PNURSP2023R358","R.G.P. 1\/177\/44"]}],"id":[{"id":"10.13039\/501100004242","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100007446","name":"King Khalid University","doi-asserted-by":"publisher","award":["PNURSP2023R358","R.G.P. 1\/177\/44"],"award-info":[{"award-number":["PNURSP2023R358","R.G.P. 1\/177\/44"]}],"id":[{"id":"10.13039\/501100007446","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>The problem of estimating the spatio-functional expectile regression for a given spatial mixing structure Xi,Yi\u2208F\u00d7R, when i\u2208ZN,N\u22651 and F is a metric space, is investigated. We have proposed the M-estimation procedure to construct the Spatial Local Linear (SLL) estimator of the expectile regression function. The main contribution of this study is the establishment of the asymptotic properties of the SLL expectile regression estimator. Precisely, we establish the almost-complete convergence with rate. This result is proven under some mild conditions on the model in the mixing framework. The implementation of the SLL estimator is evaluated using an empirical investigation. A COVID-19 data application is performed, allowing this work to highlight the substantial superiority of the SLL-expectile over SLL-quantile in risk exploration.<\/jats:p>","DOI":"10.3390\/sym15122108","type":"journal-article","created":{"date-parts":[[2023,11,23]],"date-time":"2023-11-23T11:31:25Z","timestamp":1700739085000},"page":"2108","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":1,"title":["Spatio-Functional Local Linear Asymmetric Least Square Regression Estimation: Application for Spatial Prediction of COVID-19 Propagation"],"prefix":"10.3390","volume":"15","author":[{"ORCID":"https:\/\/orcid.org\/0000-0001-6527-5783","authenticated-orcid":false,"given":"Ali","family":"Laksaci","sequence":"first","affiliation":[{"name":"Department of Mathematics, College of Science, King Khalid University, Abha 62529, Saudi Arabia"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-7801-4945","authenticated-orcid":false,"given":"Salim","family":"Bouzebda","sequence":"additional","affiliation":[{"name":"Laboratoire de Math\u00e9matiques Appliqu\u00e9es de Compi\u00e8gne (L.M.A.C.), Universit\u00e9 de Technologie de Compi\u00e8gne, 60200 Compi\u00e8gne, France"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-1306-7688","authenticated-orcid":false,"given":"Fatimah","family":"Alshahrani","sequence":"additional","affiliation":[{"name":"Department of Mathematical Sciences, College of Science, Princess Nourah bint Abdulrahman University, Riyadh 11671, Saudi Arabia"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Ouahiba","family":"Litimein","sequence":"additional","affiliation":[{"name":"Laboratory of Statistics and Stochastic Processes, University of Djillali Liabes, BP 89, Sidi Bel Abbes 22000, Algeria"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Boubaker","family":"Mechab","sequence":"additional","affiliation":[{"name":"Laboratory of Statistics and Stochastic Processes, University of Djillali Liabes, BP 89, Sidi Bel Abbes 22000, Algeria"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2023,11,23]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","unstructured":"Ripley, B.D. 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