{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,2,7]],"date-time":"2026-02-07T12:34:19Z","timestamp":1770467659995,"version":"3.49.0"},"reference-count":95,"publisher":"MDPI AG","issue":"4","license":[{"start":{"date-parts":[[2025,4,6]],"date-time":"2025-04-06T00:00:00Z","timestamp":1743897600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"Deanship of Scientific Research, Qassim University"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Entropy"],"abstract":"<jats:p>In this work, we propose a wavelet-based framework for estimating the derivatives of a density function in the setting of continuous, stationary, and ergodic processes. Our primary focus is the derivation of the integrated mean square error (IMSE) over compact subsets of Rd, which provides a quantitative measure of the estimation accuracy. In addition, a uniform convergence rate and normality are established. To establish the asymptotic behavior of the proposed estimators, we adopt a martingale approach that accommodates the ergodic nature of the underlying processes. Importantly, beyond ergodicity, our analysis does not require additional assumptions regarding the data. By demonstrating that the wavelet methodology remains valid under these weaker dependence conditions, we extend earlier results originally developed in the context of independent observations.<\/jats:p>","DOI":"10.3390\/e27040389","type":"journal-article","created":{"date-parts":[[2025,4,7]],"date-time":"2025-04-07T11:24:39Z","timestamp":1744025079000},"page":"389","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":1,"title":["Linear Wavelet-Based Estimators of Partial Derivatives of Multivariate Density Function for Stationary and Ergodic Continuous Time Processes"],"prefix":"10.3390","volume":"27","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-7630-4604","authenticated-orcid":false,"given":"Sultana","family":"Didi","sequence":"first","affiliation":[{"name":"Department of Statistics and Operations Research, College of Sciences, Qassim University, P.O. Box 6688, Buraydah 51452, 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":"LMAC (Laboratory of Applied Mathematics of Compi\u00e8gne), Universit\u00e9 de Technologie de Compi\u00e8gne, CS 60 319-60 203 Compi\u00e8gne Cedex, 60203 Compi\u00e8gne, France"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2025,4,6]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"99","DOI":"10.1111\/rssb.12111","article-title":"Non-parametric inference for density modes","volume":"78","author":"Genovese","year":"2016","journal-title":"J. R. Stat. Soc. Ser. B Stat. 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