{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T01:13:16Z","timestamp":1760058796048,"version":"build-2065373602"},"reference-count":26,"publisher":"MDPI AG","issue":"5","license":[{"start":{"date-parts":[[2025,4,26]],"date-time":"2025-04-26T00:00:00Z","timestamp":1745625600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"Scientific and Technological Innovation Programs of Higher Education Institutions in Shanxi","award":["2022L415","2022Q15"],"award-info":[{"award-number":["2022L415","2022Q15"]}]},{"name":"Shanxi Datong University Project","award":["2022L415","2022Q15"],"award-info":[{"award-number":["2022L415","2022Q15"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>The uncertain moving average model is a powerful tool to study the time series in which the data are affected by the previous disturbance terms under uncertain environments. However, the influence of uncertain disturbance terms is often ignored in the relevant statistical inference studies. In order to solve this problem, this paper constructs a symmetric statistical invariant, normal uncertain variable, by sorting out the uncertain moving average model and combining the uncertain disturbance terms, and then applies the least square principle to the parameter estimation of the symmetric statistical invariant to determine the unknown parameters and uncertain disturbance terms in the uncertain moving average model. In addition, a numerical algorithm is designed to calculate the corresponding estimators, and the problems related to hypothesis testing and forecast are also studied. Finally, a numerical example is given to illustrate the proposed method.<\/jats:p>","DOI":"10.3390\/sym17050656","type":"journal-article","created":{"date-parts":[[2025,5,2]],"date-time":"2025-05-02T11:35:13Z","timestamp":1746185713000},"page":"656","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Parameter Estimation of Uncertain Moving Average Model Based on Least Squares Principle"],"prefix":"10.3390","volume":"17","author":[{"given":"Han","family":"Wang","sequence":"first","affiliation":[{"name":"College of Mathematics and Statistics, Shanxi Datong University, Datong 037009, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-5719-8893","authenticated-orcid":false,"given":"Haiyan","family":"Shi","sequence":"additional","affiliation":[{"name":"College of Mathematics and Statistics, Shanxi Datong University, Datong 037009, China"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2025,4,26]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"267","DOI":"10.1098\/rsta.1927.0007","article-title":"On a method of investigating periodicities disturbed series, with special reference to Wolfer\u2019s sunspot numbers","volume":"226","author":"Yule","year":"1927","journal-title":"Philos. Trans. R. Soc. Lond. Ser. A"},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"518","DOI":"10.1098\/rspa.1931.0069","article-title":"On periodicity in series of related terms","volume":"131","author":"Walker","year":"1931","journal-title":"Proc. R. Soc. Lond. Ser. A"},{"key":"ref_3","unstructured":"Box, G., and Jenkins, G. (1970). Time Series Analysis: Forecasting and Control, Holden-Day."},{"key":"ref_4","doi-asserted-by":"crossref","unstructured":"Shumway, R., and Stoffer, D. (2000). Time Series Analysis and Its Applications, Springer.","DOI":"10.1007\/978-1-4757-3261-0"},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"155","DOI":"10.1007\/s10700-023-09415-5","article-title":"Solution method and parameter estimation of uncertain partial differential equation with application to China\u2019s population","volume":"23","author":"Yang","year":"2024","journal-title":"Fuzzy Optim. Decis. Mak."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"497","DOI":"10.1007\/s10700-024-09430-0","article-title":"Are the queueing systems in practice random or uncertain? Evidence from online car-hailing data in Beijing","volume":"23","author":"Liu","year":"2024","journal-title":"Fuzzy Optim. Decis. Mak."},{"key":"ref_7","unstructured":"Liu, B. (2007). Uncertainty Theory, Springer. [2nd ed.]."},{"key":"ref_8","first-page":"3","article-title":"Some research problems in uncertainty theory","volume":"3","author":"Liu","year":"2009","journal-title":"J. Uncertain Syst."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"263","DOI":"10.1007\/s10700-018-9298-z","article-title":"Uncertain time series analysis with imprecise observations","volume":"18","author":"Yang","year":"2019","journal-title":"Fuzzy Optim. Decis. Mak."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"546","DOI":"10.1080\/03081079.2020.1748616","article-title":"Prediction method of autoregressive moving average models for uncertain time series","volume":"49","author":"Lu","year":"2020","journal-title":"Int. J. Gen. Syst."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"17001","DOI":"10.1007\/s00500-020-04991-9","article-title":"Uncertain vector autoregressive model with imprecise observations","volume":"24","author":"Tang","year":"2020","journal-title":"Soft Comput."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"6915","DOI":"10.3233\/JIFS-210848","article-title":"Uncertain max-autoregressive model with imprecise observations","volume":"41","author":"Tang","year":"2021","journal-title":"J. Intell. Fuzzy Syst."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"8776","DOI":"10.1080\/03610926.2021.1906433","article-title":"Uncertain threshold autoregressive model with imprecise observations","volume":"51","author":"Tang","year":"2022","journal-title":"Commun. Stat.\u2014Theory Methods"},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"239","DOI":"10.1007\/s10700-024-09419-9","article-title":"Nonparametric uncertain time series models: Theory and application in brent crude oil spot price analysis","volume":"23","author":"Zhang","year":"2024","journal-title":"Fuzzy Optim. Decis. Mak."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"18211","DOI":"10.1007\/s00500-020-05079-0","article-title":"Least absolute deviations estimation for uncertain autoregressive model","volume":"24","author":"Yang","year":"2020","journal-title":"Soft Comput."},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"1391","DOI":"10.3233\/JIFS-201724","article-title":"Maximum likelihood estimation for uncertain autoregressive model with application to carbon dioxide emissions","volume":"40","author":"Chen","year":"2021","journal-title":"J. Intell. Fuzzy Syst."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"939","DOI":"10.1142\/S0218488520500415","article-title":"Uncertain autoregressive model via LASSO procedure","volume":"28","author":"Zhang","year":"2020","journal-title":"Int. J. Uncertain. Fuzziness Knowl.-Based Syst."},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"37","DOI":"10.1142\/S0218488521500033","article-title":"Ridge estimation for uncertain autoregressive model with imprecise observations","volume":"29","author":"Chen","year":"2021","journal-title":"Int. J. Uncertain. Fuzziness Knowl.-Based Syst."},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"2150010","DOI":"10.1142\/S1752890921500100","article-title":"Huber estimation for uncertain autoregressive model","volume":"14","author":"Liu","year":"2021","journal-title":"J. Uncertain Syst."},{"key":"ref_20","doi-asserted-by":"crossref","unstructured":"Liu, Y., and Qin, Z. (2024). Moment estimation of uncertain autoregressive model and its application in financial market. Commun. Stat.\u2014Simul. Comput.","DOI":"10.1080\/03610918.2024.2378113"},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"6649","DOI":"10.1080\/03610926.2023.2248534","article-title":"A modified uncertain maximum likelihood estimation with applications in uncertain statistics","volume":"53","author":"Liu","year":"2024","journal-title":"Commun. Stat. Theory Methods"},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"7624","DOI":"10.1080\/03610926.2023.2269451","article-title":"Estimation of uncertainty distribution function by the principle of least squares","volume":"53","author":"Liu","year":"2024","journal-title":"Commun. Stat.\u2014Theory Methods"},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"4134","DOI":"10.1080\/03610926.2020.1713373","article-title":"Least squares estimation for uncertain moving average model","volume":"50","author":"Yang","year":"2021","journal-title":"Commun. Stat.\u2014Theory Methods"},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"724","DOI":"10.1080\/03081079.2021.1950150","article-title":"Least squares estimation for the high-order uncertain moving average model with application to carbon dioxide emissions","volume":"50","author":"Xin","year":"2021","journal-title":"Int. J. Gen. Syst."},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"157","DOI":"10.1007\/s10700-021-09365-w","article-title":"Uncertain hypothesis test with application to uncertain regression analysis","volume":"21","author":"Ye","year":"2022","journal-title":"Fuzzy Optim. Decis. Mak."},{"key":"ref_26","doi-asserted-by":"crossref","unstructured":"Liu, Z., and Liu, Y. (2025). Moment estimation for uncertain moving average model with application to CO2 emission. Commun. Stat. Simul. Comput.","DOI":"10.1080\/03610918.2024.2447448"}],"container-title":["Symmetry"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2073-8994\/17\/5\/656\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,9]],"date-time":"2025-10-09T17:22:07Z","timestamp":1760030527000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2073-8994\/17\/5\/656"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2025,4,26]]},"references-count":26,"journal-issue":{"issue":"5","published-online":{"date-parts":[[2025,5]]}},"alternative-id":["sym17050656"],"URL":"https:\/\/doi.org\/10.3390\/sym17050656","relation":{},"ISSN":["2073-8994"],"issn-type":[{"type":"electronic","value":"2073-8994"}],"subject":[],"published":{"date-parts":[[2025,4,26]]}}}