{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,6,2]],"date-time":"2026-06-02T10:45:24Z","timestamp":1780397124324,"version":"3.54.1"},"reference-count":24,"publisher":"MDPI AG","issue":"2","license":[{"start":{"date-parts":[[2025,2,17]],"date-time":"2025-02-17T00:00:00Z","timestamp":1739750400000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"China University Industry-University-Research Innovation Fund","award":["2023DT001"],"award-info":[{"award-number":["2023DT001"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>In the current global context of economic integration, unexpected events have an important influence in the financial field. In 2020, the \u201cCOVID-19\u201d outbreak triggered financial turmoil throughout the whole country and even in the global market. In the wake of this era, how to sum up past developments and predict future development through change-point detection is particularly important. In this paper, four methods for detecting change-points are presented: the likelihood ratio method, least squares method, CUSUM method, and local comparison method. Considering that Bernstein polynomials have worked well in density function approximation, the multi-dimensional Bernstein polynomials are presented. The study applies multiple change-point detection methods to determine the most suitable degree of freedom mj for multi-dimensional Bernstein models, after which various rewriting expressions can be obtained. Next, \u201cCOVID-19\u201d data and money supply data are used for change-point detection with good results. Then, we focus on conducting change-point testing on the S&amp;P 500 index and SSE 50 index, indicating strong symmetry when major crisis events occur. All analyses indicate that change-point detection plays an important role in identifying major crisis events and financial shocks.<\/jats:p>","DOI":"10.3390\/sym17020302","type":"journal-article","created":{"date-parts":[[2025,2,17]],"date-time":"2025-02-17T07:48:22Z","timestamp":1739778502000},"page":"302","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":6,"title":["Study of Change-Point Detection and Applications Based on Several Statistical Methods"],"prefix":"10.3390","volume":"17","author":[{"given":"Fenglin","family":"Tian","sequence":"first","affiliation":[{"name":"School of Mathematics, Harbin Institute of Technology, Harbin 150001, China"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Yue","family":"Qi","sequence":"additional","affiliation":[{"name":"Student Affairs Department, ShanghaiTech University, Shanghai 201210, China"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Yong","family":"Wang","sequence":"additional","affiliation":[{"name":"School of Mathematics, Harbin Institute of Technology, Harbin 150001, China"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-5267-6764","authenticated-orcid":false,"given":"Boping","family":"Tian","sequence":"additional","affiliation":[{"name":"School of Mathematics, Harbin Institute of Technology, Harbin 150001, China"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"1968","published-online":{"date-parts":[[2025,2,17]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"100","DOI":"10.1093\/biomet\/41.1-2.100","article-title":"Continuous Inspection Schemes","volume":"41","author":"Page","year":"1954","journal-title":"Biometrika"},{"key":"ref_2","first-page":"18","article-title":"Overview of the latest progress in change point detection problem","volume":"40","author":"Zhang","year":"2012","journal-title":"J. Jianghan Univ. (Nat. Sci. Ed.)"},{"key":"ref_3","unstructured":"Tang, C. (2007). Statistical Inference for Change-Point Problems and Its Applications in Finance. [Doctoral Dissertation, University of Science and Technology of China]."},{"key":"ref_4","unstructured":"Li, Y. (2018). Multiple Change Points Detection Based on Binary Segementation and Its Applications in Finance. [Master\u2019s Thesis, Harbin Institute of Technology]."},{"key":"ref_5","first-page":"82","article-title":"Change Point Detection of Stock Market Liquidity in the Context of European Sovereign Debt Crisis","volume":"7","author":"Liu","year":"2014","journal-title":"J. Manag. Sci. Eng."},{"key":"ref_6","unstructured":"Cs\u00f6rg\u00f6, M., and Horv\u00e1th, L. (1997). Limit Theorems in Change-Point Analysis, John Wiley and Sons."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"149","DOI":"10.1111\/j.2517-6161.1975.tb01532.x","article-title":"Techniques for Testing the Constancy of Regression Relationships over Time","volume":"37","author":"Brown","year":"1975","journal-title":"J. R. Stat. Soc."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"385","DOI":"10.1016\/S0167-7152(98)00145-X","article-title":"Change-point in the Mean of Dependent Observations","volume":"40","author":"Kokoszka","year":"1998","journal-title":"Stat. Probab. Lett."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"445","DOI":"10.1080\/03610920008832494","article-title":"On the CUSUM Testing for Parameter Changes in GARCH(1,1) Models","volume":"29","author":"Kim","year":"2000","journal-title":"Commun. Stat.-Theory Methods"},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"173","DOI":"10.14490\/jjss.34.173","article-title":"The CUSUM Test for Parameter Change inRegression Models with ARCH Errors","volume":"34","author":"Lee","year":"2004","journal-title":"J. Jpn. Stat. Soc."},{"key":"ref_11","first-page":"69","article-title":"Asymptotic Approximations for Likeli-hood Ratio Tests and Confidence Regions for a Change-Point in the Mean of a Multivariate Normal Distribution","volume":"2","author":"James","year":"1992","journal-title":"Stat. Sin."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"72","DOI":"10.1111\/1368-423X.00102","article-title":"Critical Values for Multiple Structural Change Tests","volume":"6","author":"Bai","year":"2003","journal-title":"Econom. J."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"313","DOI":"10.1016\/j.jspi.2004.03.004","article-title":"Change Point Problems in the Model of Logistic Regression","volume":"131","author":"Gurevich","year":"2005","journal-title":"J. Stat. Plan. Inference"},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"321","DOI":"10.1016\/0167-7152(94)00190-J","article-title":"The Bernstein Polynomial Estimator of a Smooth Quantile Function","volume":"24","author":"Cheng","year":"1995","journal-title":"Stat. Probab. Lett."},{"key":"ref_15","first-page":"905","article-title":"Nonparametric Estimator Of False Discovery Rate Based On Bernstein Polynomials","volume":"18","author":"Guan","year":"2008","journal-title":"Stat. Sin."},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"61","DOI":"10.1080\/10485252.2013.827195","article-title":"On Improving Convergence Rate of Bernstein Polynomial Density Estimator","volume":"26","author":"Igarashi","year":"2014","journal-title":"J. Nonparametric Stat."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"250","DOI":"10.1080\/10485252.2016.1163349","article-title":"Efficient and Robust Density Estimation Using Bernstein Type Polynomials","volume":"28","author":"Guan","year":"2016","journal-title":"J. Nonparametric Stat."},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"831","DOI":"10.1080\/10485252.2017.1374384","article-title":"Bernstein Polynomial Model for Grouped Continuous Data","volume":"29","author":"Guan","year":"2017","journal-title":"J. Nonparametric Stat."},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"346","DOI":"10.1080\/10485252.2017.1303063","article-title":"Testing Independence Based on Bernstein Empirical Copula and Copula Density","volume":"29","author":"Belalia","year":"2017","journal-title":"J. Nonparametric Stat."},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"233","DOI":"10.1007\/BF01895321","article-title":"Two-Dimensional Bernstein Polynomial Density Estimators","volume":"41","author":"Tenbusch","year":"1994","journal-title":"Metrika"},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"321","DOI":"10.1080\/02331888.2019.1574299","article-title":"Bernstein Polynomial Model for Nonparametric Multivariate Density","volume":"53","author":"Wang","year":"2019","journal-title":"Statistics"},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"239","DOI":"10.1007\/BF01398235","article-title":"The Degree of Approximation by Polynomials with Positive Coefficients","volume":"151","author":"Lorentz","year":"1963","journal-title":"Math. Ann."},{"key":"ref_23","unstructured":"Cheney, E.W. (1982). Introduction to Approximation Theory, AMS Chelsea Publishing. [2nd ed.]."},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"195","DOI":"10.1137\/1026034","article-title":"Mixture Densities, Maximum Likelihood and the EM Algorithm","volume":"26","author":"Redner","year":"1984","journal-title":"SIAM Rev."}],"container-title":["Symmetry"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2073-8994\/17\/2\/302\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,9]],"date-time":"2025-10-09T16:36:02Z","timestamp":1760027762000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2073-8994\/17\/2\/302"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2025,2,17]]},"references-count":24,"journal-issue":{"issue":"2","published-online":{"date-parts":[[2025,2]]}},"alternative-id":["sym17020302"],"URL":"https:\/\/doi.org\/10.3390\/sym17020302","relation":{},"ISSN":["2073-8994"],"issn-type":[{"value":"2073-8994","type":"electronic"}],"subject":[],"published":{"date-parts":[[2025,2,17]]}}}