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In this paper, a computational approach, based on the non-parametric methods, is used to investigate the similarities, and to classify several linear and non-linear regression models with symmetric errors. The ability of each given approach is then evaluated using simulated and real world practical datasets.<\/jats:p>","DOI":"10.3390\/sym11060820","type":"journal-article","created":{"date-parts":[[2019,6,20]],"date-time":"2019-06-20T10:49:59Z","timestamp":1561027799000},"page":"820","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":47,"title":["On Comparing and Classifying Several Independent Linear and Non-Linear Regression Models with Symmetric Errors"],"prefix":"10.3390","volume":"11","author":[{"given":"Ji-Jun","family":"Pan","sequence":"first","affiliation":[{"name":"College of Mathematics, Dianxi Science and Technology, Normal University, Lincang 677000, China"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-4686-0930","authenticated-orcid":false,"given":"Mohammad Reza","family":"Mahmoudi","sequence":"additional","affiliation":[{"name":"Department of Statistics, Faculty of Science, Fasa University, Fasa 74616 86131, Iran"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-0286-7244","authenticated-orcid":false,"given":"Dumitru","family":"Baleanu","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Faculty of Art and Sciences, Cankaya University Balgat, Ankara 06530, Turkey"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-2774-2464","authenticated-orcid":false,"given":"Mohsen","family":"Maleki","sequence":"additional","affiliation":[{"name":"Department of Statistics, Faculty of Science, Shiraz University, Shiraz 71946 85115, Iran"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"1968","published-online":{"date-parts":[[2019,6,20]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","unstructured":"Wan, J., Zhang, D., Xu, W., and Guo, Q. (2019). Parameter Estimation of Multi Frequency Hopping Signals Based on Space-Time-Frequency Distribution. Symmetry, 11.","DOI":"10.3390\/sym11050648"},{"key":"ref_2","doi-asserted-by":"crossref","unstructured":"Sajid, M., Shafique, T., Riaz, I., Imran, M., Jabbar Aziz Baig, M., Baig, S., and Manzoor, S. (2018). Facial asymmetry-based anthropometric differences between gender and ethnicity. Symmetry, 10.","DOI":"10.3390\/sym10070232"},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"665","DOI":"10.1007\/s40995-017-0288-8","article-title":"Testing the Difference between Two Independent Time Series Models","volume":"41","author":"Mahmouudi","year":"2017","journal-title":"Iran. J. Sci. Technol. Trans. A Sci."},{"key":"ref_4","first-page":"3","article-title":"On the Probable Error of a Coefficient of Correlation Deduced from a Small Sample","volume":"1","author":"Fisher","year":"1921","journal-title":"Metron"},{"key":"ref_5","first-page":"71","article-title":"Inferrence on the Ratio of Correlations of Two Independent Populations","volume":"7","author":"Mahmoudi","year":"2014","journal-title":"J. Math. Ext."},{"key":"ref_6","unstructured":"Howell, D.C. (2007). Statistical Methods for Psychology, Thomson Wadsworth. [6th ed.]."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"271","DOI":"10.1214\/aoms\/1177731867","article-title":"The Selection of Variates for Use in Prediction with Some Comments on the General Problem of Nuisance Parameters","volume":"11","author":"Hotelling","year":"1940","journal-title":"Ann. Math. Stat."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"396","DOI":"10.1111\/j.2517-6161.1959.tb00346.x","article-title":"The Comparison of Regression Variables","volume":"21","author":"Williams","year":"1959","journal-title":"J. R. Stat. Soc. Ser. B"},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"245","DOI":"10.1037\/0033-2909.87.2.245","article-title":"Tests for Comparing Elements of a Correlation Matrix","volume":"87","author":"Steiger","year":"1980","journal-title":"Psychol. Bull."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"172","DOI":"10.1037\/0033-2909.111.1.172","article-title":"Comparing Correlated Correlation Coefficients","volume":"111","author":"Meng","year":"1992","journal-title":"Psychol. Bull."},{"key":"ref_11","doi-asserted-by":"crossref","unstructured":"Peter, C.C., and Van Voorhis, W.R. (1940). Statistical Procedures and Their Mathematical Bases, McGraw-Hill.","DOI":"10.1037\/13596-000"},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"178","DOI":"10.1037\/1082-989X.1.2.178","article-title":"Comparing Correlated but Nonoverlapping Correlations","volume":"1","author":"Raghunathan","year":"1996","journal-title":"Psychol. Methods"},{"key":"ref_13","first-page":"395","article-title":"Multiple Comparison of Several Linear Regression Lines","volume":"99","author":"Liu","year":"2004","journal-title":"J. R. Stat. Soc. Ser. B"},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"144","DOI":"10.1002\/bimj.200610322","article-title":"Operability Region Equivalence: Simultaneous Confidence Bands for the Equivalence of Two Regression Models Over Restricted Regions","volume":"49","author":"Liu","year":"2007","journal-title":"Biom. J."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"2759","DOI":"10.1002\/sim.2756","article-title":"Pooling Batches in Drug Stability Study by Using Constant-width Simultaneous Confidence Bands","volume":"26","author":"Liu","year":"2007","journal-title":"Stat. Med."},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"57","DOI":"10.1016\/j.jspi.2005.09.007","article-title":"Some New Methods for the Comparison of Two Linear Regression Models","volume":"137","author":"Liu","year":"2007","journal-title":"J. Stat. Plan. Inference"},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"1213","DOI":"10.1016\/j.jspi.2006.02.004","article-title":"Easy-to-Construct Confidence Bands for Comparing Two Simple Linear Regression Lines","volume":"137","author":"Hayter","year":"2007","journal-title":"J. Stat. Plan. Inference"},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"1475","DOI":"10.1016\/j.csda.2010.01.022","article-title":"Simultaneous Confidence Bands for all Contrasts of Three or More Simple Linear Regression Models over an Interval","volume":"54","author":"Jamshidian","year":"2010","journal-title":"Comput. Stat Data Anal."},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"1453","DOI":"10.1007\/s00180-016-0703-1","article-title":"Testing the equality of several linear regression models","volume":"32","author":"Marques","year":"2016","journal-title":"Comput. Stat."},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"6284","DOI":"10.1080\/03610926.2014.960584","article-title":"Testing the Difference between Two Independent Regression Models","volume":"45","author":"Mahmoudi","year":"2016","journal-title":"Commun. Stat. Theory Methods"},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"449","DOI":"10.1520\/JTE20170461","article-title":"On Comparing Two Dependent Linear and Nonlinear Regression Models","volume":"47","author":"Mahmoudi","year":"2018","journal-title":"J. Test. Eval."},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"2919","DOI":"10.1080\/03610926.2017.1343847","article-title":"Testing the Equality of Two Independent Regression Models","volume":"47","author":"Mahmoudi","year":"2018","journal-title":"Commun. Stat. Theory Methods"},{"key":"ref_23","unstructured":"Conover, W.J. (1980). Practical Nonparametric Statistics, John Wiley. [3rd ed.]."},{"key":"ref_24","first-page":"675","article-title":"The Use of Ranks to Avoid the Assumption of Normality Implicit in the Analysis of Variance","volume":"32","author":"Friedman","year":"1937","journal-title":"J. R. Stat. Soc. Ser. B"},{"key":"ref_25","first-page":"109","article-title":"A Correction: The Use of Ranks to Avoid the Assumption of Normality Implicit in the Analysis of Variance","volume":"34","author":"Friedman","year":"1939","journal-title":"J. R. Stat. Soc. Ser. B"},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"86","DOI":"10.1214\/aoms\/1177731944","article-title":"A Comparison of Alternative Tests of Significance for the Problem of m Rankings","volume":"11","author":"Friedman","year":"1940","journal-title":"Ann. Math. 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