{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T00:57:57Z","timestamp":1760144277695,"version":"build-2065373602"},"reference-count":15,"publisher":"MDPI AG","issue":"4","license":[{"start":{"date-parts":[[2024,3,31]],"date-time":"2024-03-31T00:00:00Z","timestamp":1711843200000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>This paper focuses on the estimation and simulation of a specific subset of bilinear time series models characterized by dynamic exponential coefficients. Employing an exponential framework, we delve into the implications of the exponential function for our estimation process. Our primary aim is to estimate the coefficients of the proposed model using exponential coefficients derived from time-varying parameters. Through this investigation, our goal is to shed light on the asymptotic behaviors of the estimators and scrutinize their existence and probabilistic traits, drawing upon the foundational theorem established by Klimko and Nilsen. The least squares approach is pivotal in both estimating coefficients and analyzing estimator behavior. Moreover, we present a practical application to underscore the real-world implications of our research. By offering concrete examples of applications and simulations, we endeavor to provide readers with a comprehensive understanding of the implications of our work within the realm of time series analysis, specifically focusing on bilinear models and time-varying exponential coefficients. This multifaceted approach underscores the potential impact and practical relevance of our findings, contributing to the advancement of the field of time series analysis. To discern the symmetry characteristics of the model, we estimate it using coefficients that sum to zero and conduct a brief comparative analysis of two bilinear models.<\/jats:p>","DOI":"10.3390\/sym16040405","type":"journal-article","created":{"date-parts":[[2024,3,31]],"date-time":"2024-03-31T13:42:30Z","timestamp":1711892550000},"page":"405","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Novel Insights into Estimation of Bilinear Time Series Models with Exponential and Symmetric Coefficients"],"prefix":"10.3390","volume":"16","author":[{"given":"Mamon","family":"Abu Hammad","sequence":"first","affiliation":[{"name":"Department of Mathematics, Al-Zaytoonah University of Jordan, Amman 11733, Jordan"}]},{"given":"Nabil","family":"Laiche","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Seville University, 41012 Seville, Spain"},{"name":"Department of Mathematics, University of Oum El Bouaghi, Oum El Bouaghi 04000, Algeria"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-1145-0256","authenticated-orcid":false,"given":"Omar","family":"Alomari","sequence":"additional","affiliation":[{"name":"College of Engineering and Technology, American University of the Middle East, Egaila 54200, Kuwait"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-7023-5719","authenticated-orcid":false,"given":"Huthaifa","family":"Abuhammad","sequence":"additional","affiliation":[{"name":"Department of Computer Science, Middle East University, Amman 11831, Jordan"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-3313-506X","authenticated-orcid":false,"given":"Shameseddin","family":"Alshorm","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Al-Zaytoonah University of Jordan, Amman 11733, Jordan"}]}],"member":"1968","published-online":{"date-parts":[[2024,3,31]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"355","DOI":"10.1081\/SAP-120028595","article-title":"Estimation of some bilinear time series models with time varying coefficients","volume":"22","author":"Bibi","year":"2004","journal-title":"Stochastic Analysis Appl."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"1070","DOI":"10.1080\/03610918.2017.1406507","article-title":"Moment method estimation of first-order continuous-time bilinear processes","volume":"48","author":"Bibi","year":"2019","journal-title":"Commun. Stat.-Simul. Comput."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"1523","DOI":"10.1360\/02ys9164","article-title":"Stationary solution and parametric estimation for bilinear model driven by ARCH noises","volume":"45","author":"Pan","year":"2002","journal-title":"Sci. China Ser. A Math."},{"key":"ref_4","doi-asserted-by":"crossref","unstructured":"Xekalaki, E., and Degiannakis, S. (2010). ARCH Models for Financial Applications, John Wiley & Sons.","DOI":"10.1002\/9780470688014"},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"453","DOI":"10.18576\/jsap\/100215","article-title":"A Comparative Study Based on Exponential Progressive Type II Censored Data","volume":"10","author":"AAlmutlak","year":"2021","journal-title":"J. Stat. Appl. Probab."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"1675","DOI":"10.29294\/IJASE.7.2.2020.1675-1690","article-title":"On non-parametric criteria for random communication and processes relationship","volume":"7","author":"Farah","year":"2020","journal-title":"Int. J. Adv. Sci. Eng."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"6411","DOI":"10.3934\/math.2024312","article-title":"Theoretical analysis of a class of \u2014Caputo fractional differential equations in Banach space","volume":"9","author":"Hammad","year":"2024","journal-title":"Aims Math."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"4046","DOI":"10.1080\/03610926.2019.1594300","article-title":"Yule-Walker type estimator of first-order time-varying periodic bilinear differential model for stochastic processes","volume":"49","author":"Bibi","year":"2020","journal-title":"Commun. 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An Introduction to Bispectral Analysis and Bilinear Time Series Models, Springer Science & Business Media."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"629","DOI":"10.1214\/aos\/1176344207","article-title":"On conditional least squares estimation for stochastic processes","volume":"6","author":"Klimko","year":"1978","journal-title":"Ann. Stat."},{"key":"ref_14","first-page":"771","article-title":"Estimating continuous-time bilinear process with moments method","volume":"30","author":"Bibi","year":"2023","journal-title":"Nonlinear Stud."},{"key":"ref_15","doi-asserted-by":"crossref","unstructured":"Francq, C., and Zakoian, J.M. (2019). GARCH Models: Structure, Statistical Inference and Financial Applications, John Wiley & Sons.","DOI":"10.1002\/9781119313472"}],"container-title":["Symmetry"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2073-8994\/16\/4\/405\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T14:21:42Z","timestamp":1760106102000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2073-8994\/16\/4\/405"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2024,3,31]]},"references-count":15,"journal-issue":{"issue":"4","published-online":{"date-parts":[[2024,4]]}},"alternative-id":["sym16040405"],"URL":"https:\/\/doi.org\/10.3390\/sym16040405","relation":{},"ISSN":["2073-8994"],"issn-type":[{"type":"electronic","value":"2073-8994"}],"subject":[],"published":{"date-parts":[[2024,3,31]]}}}