{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T02:27:23Z","timestamp":1760149643205,"version":"build-2065373602"},"reference-count":35,"publisher":"MDPI AG","issue":"8","license":[{"start":{"date-parts":[[2023,8,13]],"date-time":"2023-08-13T00:00:00Z","timestamp":1691884800000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"Princess Nourah bint Abdulrahman University","award":["PNURSP2023R443","PSAU\/2023\/R\/1444"],"award-info":[{"award-number":["PNURSP2023R443","PSAU\/2023\/R\/1444"]}]},{"name":"Prince Sattam bin Abdulaziz Universities","award":["PNURSP2023R443","PSAU\/2023\/R\/1444"],"award-info":[{"award-number":["PNURSP2023R443","PSAU\/2023\/R\/1444"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>Several models for time series with integer values have been published as a result of the substantial demand for the description of process stability having discrete marginal distributions. One of these models is the mixed count geometric integer autoregressive of order one (MCGINAR(1)), which is based on two thinning operators. This study examines how the estimates of the spectral density functions of the MCGINAR(1) model are affected by fuzzy time series Markov chain (FTSMC). Regarding this study\u2019s context, the higher-order moments, central moments and spectral density functions of MCGINAR(1) are computed. The anticipated realizations of the generated realizations for this model are obtained based on FTSMC. In the case of generated and anticipated realizations, several lag windows are used to smooth the spectral density estimators. The generated realization estimates are compared with the anticipated realization estimates using the MSE to ascertain the FTSMC\u2019s role in improving the estimation process.<\/jats:p>","DOI":"10.3390\/sym15081577","type":"journal-article","created":{"date-parts":[[2023,8,14]],"date-time":"2023-08-14T10:20:14Z","timestamp":1692008414000},"page":"1577","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":1,"title":["A New Statistical Technique to Enhance MCGINAR(1) Process Estimates under Symmetric and Asymmetric Data: Fuzzy Time Series Markov Chain and Its Characteristics"],"prefix":"10.3390","volume":"15","author":[{"given":"Mohammed H.","family":"El-Menshawy","sequence":"first","affiliation":[{"name":"Department of Mathematics, Faculty of Science, Al-Azhar University, Nasr City, Cairo 11884, Egypt"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Abd El-Moneim A. M.","family":"Teamah","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Faculty of Science, Tanta University, Tanta 31527, Egypt"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-5619-210X","authenticated-orcid":false,"given":"Mohamed S.","family":"Eliwa","sequence":"additional","affiliation":[{"name":"Department of Statistics and Operation Research, College of Science, Qassim University, Buraydah 51482, Saudi Arabia"},{"name":"Department of Mathematics, Faculty of Science, Mansoura University, Mansoura 35516, Egypt"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Laila A.","family":"Al-Essa","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\/0000-0002-7585-5519","authenticated-orcid":false,"given":"Mahmoud","family":"El-Morshedy","sequence":"additional","affiliation":[{"name":"Department of Mathematics, College of Science and Humanities in Al-Kharj, Prince Sattam bin Abdulaziz University, Al-Kharj 11942, Saudi Arabia"},{"name":"Department of Statistics and Computer Science, Faculty of Science, Mansoura University, Mansoura 35516, Egypt"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-7828-3536","authenticated-orcid":false,"given":"Rashad M.","family":"EL-Sagheer","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Faculty of Science, Al-Azhar University, Nasr City, Cairo 11884, Egypt"},{"name":"High Institute of Computer and Management Information System, First Statement, New Cairo, Cairo 11865, Egypt"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2023,8,13]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"269","DOI":"10.1016\/0165-0114(93)90372-O","article-title":"Fuzzy time series and its models","volume":"54","author":"Song","year":"1993","journal-title":"Fuzzy Sets Syst."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"80236","DOI":"10.1109\/ACCESS.2021.3084048","article-title":"A novel stochastic fuzzy time series forecasting model based on a new partition method","volume":"9","author":"Alyousifi","year":"2021","journal-title":"IEEE Access"},{"key":"ref_3","doi-asserted-by":"crossref","unstructured":"Alyousifi, Y., Othman, M., Sokkalingam, R., Faye, I., and Silva, P.C. 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