{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,11,13]],"date-time":"2025-11-13T12:48:00Z","timestamp":1763038080948,"version":"build-2065373602"},"reference-count":35,"publisher":"MDPI AG","issue":"6","license":[{"start":{"date-parts":[[2025,5,24]],"date-time":"2025-05-24T00:00:00Z","timestamp":1748044800000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"Princess Nourah bint Abdulrahman University Researchers","award":["PNURSP2025R515","RGP1\/41\/46"],"award-info":[{"award-number":["PNURSP2025R515","RGP1\/41\/46"]}]},{"DOI":"10.13039\/501100004242","name":"Princess Nourah bint Abdulrahman University","doi-asserted-by":"publisher","award":["PNURSP2025R515","RGP1\/41\/46"],"award-info":[{"award-number":["PNURSP2025R515","RGP1\/41\/46"]}],"id":[{"id":"10.13039\/501100004242","id-type":"DOI","asserted-by":"publisher"}]},{"name":"Deanship of Scientific Research and Graduate Studies at King Khalid University","award":["PNURSP2025R515","RGP1\/41\/46"],"award-info":[{"award-number":["PNURSP2025R515","RGP1\/41\/46"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Entropy"],"abstract":"<jats:p>In this paper, we investigate the recursive L1 estimator of the conditional mode when the input variable takes values in a pseudo-metric space. The new proposed estimator is constructed under an ergodicity assumption, which provides a robust alternative to the standard mixing processes in various practical settings. The particular interest of this contribution arises from the difficulty in incorporating the mathematical properties of a functional mixing process. In contrast, ergodicity is characterized by the Kolmogorov\u2013Sinai entropy, which measures the dynamics, the sparsity, and the microscopic fluctuations of the functional process. Using an observation sampled from ergodic functional time series (fts), we establish the asymptotic properties of this estimator. In particular, we derive its convergence rate and show Borel\u2013Cantelli (BC) consistency. The general expression for the convergence rate is then specialized to several notable scenarios, including the independence case, the classical kernel method, and the vector-valued case. Finally, numerical experiments on both simulated and real-world datasets demonstrate the superiority of the L1-recursive estimator compared to existing competitors.<\/jats:p>","DOI":"10.3390\/e27060552","type":"journal-article","created":{"date-parts":[[2025,5,25]],"date-time":"2025-05-25T20:26:50Z","timestamp":1748204810000},"page":"552","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":2,"title":["Scalar-on-Function Mode Estimation Using Entropy and Ergodic Properties of Functional Time Series Data"],"prefix":"10.3390","volume":"27","author":[{"ORCID":"https:\/\/orcid.org\/0009-0009-3860-6917","authenticated-orcid":false,"given":"Mohammed B.","family":"Alamari","sequence":"first","affiliation":[{"name":"Department of Mathematics, College of Science, King Khalid University, Abha 62223, Saudi Arabia"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-9198-4903","authenticated-orcid":false,"given":"Fatimah A.","family":"Almulhim","sequence":"additional","affiliation":[{"name":"Department of Mathematical Sciences, College of Science, Princess Nourah bint Abdulrahman University, P.O. Box 84428, Riyadh 11671, Saudi Arabia"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-4651-3210","authenticated-orcid":false,"given":"Ibrahim M.","family":"Almanjahie","sequence":"additional","affiliation":[{"name":"Department of Mathematics, College of Science, King Khalid University, Abha 62223, Saudi Arabia"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-7801-4945","authenticated-orcid":false,"given":"Salim","family":"Bouzebda","sequence":"additional","affiliation":[{"name":"Universit\u00e9 de Technologie de Compi\u00e8gne, LMAC (Laboratory of Applied Mathematics of Compi\u00e8gne), 60203 Compi\u00e8gne, France"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-6527-5783","authenticated-orcid":false,"given":"Ali","family":"Laksaci","sequence":"additional","affiliation":[{"name":"Department of Mathematics, College of Science, King Khalid University, Abha 62223, Saudi Arabia"}]}],"member":"1968","published-online":{"date-parts":[[2025,5,24]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"227","DOI":"10.1016\/0378-3758(86)90099-6","article-title":"A note on prediction via estimation of the conditional mode function","volume":"15","author":"Collomb","year":"1986","journal-title":"J. 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