{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,11,2]],"date-time":"2025-11-02T05:49:34Z","timestamp":1762062574142,"version":"build-2065373602"},"reference-count":37,"publisher":"MDPI AG","issue":"9","license":[{"start":{"date-parts":[[2022,8,24]],"date-time":"2022-08-24T00:00:00Z","timestamp":1661299200000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/501100007446","name":"King Khalid University","doi-asserted-by":"publisher","award":["RGP.2\/34\/43"],"award-info":[{"award-number":["RGP.2\/34\/43"]}],"id":[{"id":"10.13039\/501100007446","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Axioms"],"abstract":"<jats:p>In this paper, the effect of fuzzy time series on estimates of the spectral, bispectral and normalized bispectral density functions are studied. This study is conducted for one of the integer autoregressive of order one (INAR(1)) models. The model of interest here is the dependent counting geometric INAR(1) which is symbolized by (DCGINAR(1)). A realization is generated for this model of size n = 500 for estimation. Based on fuzzy time series, the forecasted observations of this model are obtained. The estimators of spectral, bispectral and normalized bispectral density functions are smoothed by different one- and two-dimensional lag windows. Finally, after the smoothing, all estimators are studied in the case of generated and forecasted observations of the DCGINAR(1) model. We investigate the contribution of the fuzzy time series to the smoothing of these estimates through the results.<\/jats:p>","DOI":"10.3390\/axioms11090423","type":"journal-article","created":{"date-parts":[[2022,8,24]],"date-time":"2022-08-24T21:03:51Z","timestamp":1661375031000},"page":"423","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":3,"title":["Effect of Fuzzy Time Series on Smoothing Estimation of the INAR(1) Process"],"prefix":"10.3390","volume":"11","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-7585-5519","authenticated-orcid":false,"given":"Mahmoud","family":"El-Morshedy","sequence":"first","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 Mathematics, Faculty of Science, Mansoura University, Mansoura 35516, Egypt"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Mohammed H.","family":"El-Menshawy","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Faculty of Science, Al-Azhar University, Nasr City 11884, Egypt"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Mohammed M. A.","family":"Almazah","sequence":"additional","affiliation":[{"name":"Department of Mathematics, College of Sciences and Arts (Muhyil), King Khalid University, Muhyil 61421, Saudi Arabia"},{"name":"Department of Mathematics and Computer, College of Sciences, Ibb University, Ibb 70270, Yemen"}],"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 11884, Egypt"},{"name":"High Institute of Computer and Management Information System, First Statement, New Cairo 11865, 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, P.O. Box 6644, Buraydah 51482, Saudi Arabia"},{"name":"Department of Mathematics, International Telematic University Uninettuno, I-00186 Rome, Italy"},{"name":"Department of Statistics and Computer Sciences, Faculty of Science, Mansoura University, Mansoura 35516, Egypt"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2022,8,24]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"645","DOI":"10.1111\/j.1752-1688.1985.tb05379.x","article-title":"Some simple models for discrete variate time series 1","volume":"21","author":"McKenzie","year":"1985","journal-title":"JAWRA J. Am. Water Resour. Assoc."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"261","DOI":"10.1111\/j.1467-9892.1987.tb00438.x","article-title":"First-order integer-valued autoregressive (INAR(1)) process","volume":"8","author":"Alzaid","year":"1987","journal-title":"J. Time Ser. 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