{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,12,20]],"date-time":"2025-12-20T22:13:08Z","timestamp":1766268788440,"version":"build-2065373602"},"reference-count":72,"publisher":"MDPI AG","issue":"7","license":[{"start":{"date-parts":[[2023,7,24]],"date-time":"2023-07-24T00:00:00Z","timestamp":1690156800000},"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 Researchers Supporting Project","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"}]},{"name":"The Deanship of Scientific Research at King Khalid University through the Research Groups Program","award":["PNURSP2023R358","R.G.P. 1\/177\/44"],"award-info":[{"award-number":["PNURSP2023R358","R.G.P. 1\/177\/44"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Entropy"],"abstract":"<jats:p>The convergence rate for free-distribution functional data analyses is challenging. It requires some advanced pure mathematics functional analysis tools. This paper aims to bring several contributions to the existing functional data analysis literature. First, we prove in this work that Kolmogorov entropy is a fundamental tool in characterizing the convergence rate of the local linear estimation. Precisely, we use this tool to derive the uniform convergence rate of the local linear estimation of the conditional cumulative distribution function and the local linear estimation conditional quantile function. Second, a central limit theorem for the proposed estimators is established. These results are proved under general assumptions, allowing for the incomplete functional time series case to be covered. Specifically, we model the correlation using the ergodic assumption and assume that the response variable is collected with missing at random. Finally, we conduct Monte Carlo simulations to assess the finite sample performance of the proposed estimators.<\/jats:p>","DOI":"10.3390\/e25071108","type":"journal-article","created":{"date-parts":[[2023,7,25]],"date-time":"2023-07-25T01:21:27Z","timestamp":1690248087000},"page":"1108","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":3,"title":["Kolmogorov Entropy for Convergence Rate in Incomplete Functional Time Series: Application to Percentile and Cumulative Estimation in High Dimensional Data"],"prefix":"10.3390","volume":"25","author":[{"given":"Ouahiba","family":"Litimein","sequence":"first","affiliation":[{"name":"Laboratory of Statistics and Stochastic Processes, University of Djillali Liabes, BP 89, Sidi Bel Abbes 22000, Algeria"}],"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, P.O. Box 84428, Riyadh 11671, 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\/0000-0001-6527-5783","authenticated-orcid":false,"given":"Ali","family":"Laksaci","sequence":"additional","affiliation":[{"name":"Department of Mathematics, College of Science, King Khalid University, Abha 62529, Saudi Arabia"}],"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":[[2023,7,24]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","unstructured":"Ramsay, J.O., and Silverman, B.W. 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