{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T01:36:01Z","timestamp":1760146561487,"version":"build-2065373602"},"reference-count":28,"publisher":"MDPI AG","issue":"11","license":[{"start":{"date-parts":[[2024,11,18]],"date-time":"2024-11-18T00:00:00Z","timestamp":1731888000000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"National Natural Science Foundation of China","award":["11701368"],"award-info":[{"award-number":["11701368"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>This paper considers a semi-parametric errors-in-variables (EV) model, \u03b7i=xi\u03b2+g(\u03c4i)+\u03f5i, \u03bei=xi+\u03b4i, 1\u2a7di\u2a7dn. The properties of estimators are investigated under conditions of missing data and strong mixing errors. Three approaches are used to handle missing data: direct deletion, imputation, and the regression surrogate. Furthermore, estimators for the coefficient \u03b2 and the nonparametric function g(\u00b7) are obtained. Notably, both estimators achieve strong consistency at a rate of o(n\u22121\/4), exhibiting a symmetry in their convergence rates, and they also demonstrate asymptotic normality. Additionally, the validity of our theoretical results is supported by simulations demonstrating the finite sample behaviour of these estimators.<\/jats:p>","DOI":"10.3390\/sym16111544","type":"journal-article","created":{"date-parts":[[2024,11,18]],"date-time":"2024-11-18T13:03:14Z","timestamp":1731934994000},"page":"1544","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Symmetrical Convergence Rates and Asymptotic Properties of Estimators in a Semi-Parametric Errors-in-Variables Model with Strong Mixing Errors and Missing Responses"],"prefix":"10.3390","volume":"16","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-8419-5655","authenticated-orcid":false,"given":"Jingjing","family":"Zhang","sequence":"first","affiliation":[{"name":"College of Science, University of Shanghai for Science and Technology, Shanghai 200093, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Haiqin","family":"Yan","sequence":"additional","affiliation":[{"name":"College of Science, University of Shanghai for Science and Technology, Shanghai 200093, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Tingting","family":"Hu","sequence":"additional","affiliation":[{"name":"United Automotive Electronics Co., Ltd., Shanghai 201206, China"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2024,11,18]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"310","DOI":"10.1080\/01621459.1986.10478274","article-title":"Semiparametric estimates of the relation between weather and electricity sales","volume":"81","author":"Engle","year":"1986","journal-title":"J. Am. Stat. Assoc."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"1130","DOI":"10.1080\/00949655.2016.1249482","article-title":"Least-trimmed squares: Asymptotic normality of robust estimator in semiparametric regression models","volume":"87","author":"Roozbeh","year":"2017","journal-title":"J. Stat. Comput. Simul."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"1212","DOI":"10.1007\/s11424-020-8210-4","article-title":"Asymptotic normality for wavelet estimators in heteroscedastic semi-parametric model with random errors","volume":"33","author":"Ding","year":"2020","journal-title":"J. Syst. Sci. Complex."},{"key":"ref_4","first-page":"378","article-title":"Error variance estimation in partially linear varying coefficient models","volume":"21","author":"Wei","year":"2008","journal-title":"Math. Appl."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"61","DOI":"10.1186\/s13660-019-2017-7","article-title":"Difference-based M-estimator of generalized semiparametric model with NSD errors","volume":"2019","author":"Fu","year":"2019","journal-title":"J. Inequal. Appl."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"109","DOI":"10.1016\/0304-4076(85)90134-4","article-title":"Panel data from time series of cross-sections","volume":"30","author":"Deaton","year":"1985","journal-title":"J. Econom."},{"key":"ref_7","doi-asserted-by":"crossref","unstructured":"Chen, P., Kong, N., and Sung, S.H. (2016). Complete convergence for weighted sums of i.i.d. random variables with applications in regression estimation and EV model. Commun. Stat., 3599\u20133613.","DOI":"10.1080\/03610926.2015.1066817"},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"1079","DOI":"10.1016\/j.jmva.2010.01.003","article-title":"Statistical inference for panel data semiparametric partially linear regression models with heteroscedastic errors","volume":"101","author":"You","year":"2010","journal-title":"J. Multiv. Anal."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"165","DOI":"10.1007\/s10182-013-0224-y","article-title":"Asymptotic normality of estimators in heteroscedastic errors-in-variables model","volume":"98","author":"Zhang","year":"2014","journal-title":"AStA Adv. Stat. Anal."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"127","DOI":"10.1016\/j.laa.2018.04.016","article-title":"Ridge estimation in semiparametric linear measurement error models","volume":"552","author":"Emami","year":"2018","journal-title":"Lin. Algebra Appl."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"2263","DOI":"10.1016\/j.jspi.2009.02.021","article-title":"Moderate deviations for LS estimator in simple linear EV regression model","volume":"139","author":"Miao","year":"2009","journal-title":"J. Stat. Plan. Inference"},{"key":"ref_12","first-page":"600","article-title":"Strong laws for weighted sums of -mixing random variables and applications in errors-in-variables regression models","volume":"26","author":"Hu","year":"2017","journal-title":"Stat. Methods Appl."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"930","DOI":"10.1515\/math-2020-0052","article-title":"Asymptotic normality and mean consistency of LS estimators in the errors-in-variables model with dependent errors","volume":"18","author":"Zhang","year":"2020","journal-title":"Open Math."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"271","DOI":"10.1080\/02331888.2022.2038166","article-title":"Estimation for a hybrid model of functional and linear measurement errors regression with missing response","volume":"56","author":"Zou","year":"2022","journal-title":"Statistics"},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"1637","DOI":"10.1007\/s00180-020-00967-3","article-title":"Estimation in partially linear varying-coefficient errors-in-variables models with missing response variables","volume":"35","author":"Xiao","year":"2020","journal-title":"Comput. Stat."},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"1141022","DOI":"10.1155\/2021\/1141022","article-title":"Statistical inference for the heteroscedastic partially linear varying-coefficient errors-in-variables model with missing censoring indicators","volume":"2021","author":"Zou","year":"2021","journal-title":"Discret. Dyn. Nat. Soc."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"1232","DOI":"10.1080\/02331888.2020.1867857","article-title":"Asymptotic properties for the estimators in heteroscedastic semiparametric EV models with \u03b1-mixing errors","volume":"54","author":"Xi","year":"2021","journal-title":"Statistics"},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"4862820","DOI":"10.1155\/2022\/4862820","article-title":"Asymptotic properties for estimators in a semiparametric EV model with NA errors and missing responses","volume":"2022","author":"Zhang","year":"2022","journal-title":"Discret. Dyn. Nat. Soc."},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"2547329","DOI":"10.1155\/2023\/2547329","article-title":"Statistical inference for estimators in a semiparametric EV model with linear process errors and missing responses","volume":"2023","author":"Zhang","year":"2023","journal-title":"Math. Probl. Eng."},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1016\/j.jmva.2008.03.006","article-title":"Berry-Esseen type bounds of estimators in a semiparametric model with linear process errors","volume":"100","author":"Liang","year":"2009","journal-title":"J. Multiv. Anal."},{"key":"ref_21","first-page":"883","article-title":"Convergence for weighted sums of negatively associated random variables","volume":"41","author":"Liang","year":"2004","journal-title":"Stat. Probab. Lett."},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"1470","DOI":"10.1016\/j.jmva.2006.10.003","article-title":"Estimation in partially linear models with missing responses at random","volume":"98","author":"Wang","year":"2007","journal-title":"J. Multiv. Anal."},{"key":"ref_23","doi-asserted-by":"crossref","unstructured":"H\u00e4rdle, W., Liang, H., and Gao, J.T. (2000). Partial Linear Models, Physica-Verlag.","DOI":"10.1007\/978-3-642-57700-0"},{"key":"ref_24","first-page":"1025","article-title":"The convergence of the weighted sum for strong mixing dependent variable and its application","volume":"45","author":"Xu","year":"2002","journal-title":"J. Math."},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"178","DOI":"10.1137\/1104015","article-title":"Some limit theorems for random functions","volume":"4","author":"Volkonskii","year":"1959","journal-title":"Theory Probab."},{"key":"ref_26","unstructured":"Hall, P., and Heyde, C.C. (1980). Martingale Limit Theory and Its Applications, Academic Press."},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"2263","DOI":"10.1080\/03610920701215266","article-title":"The central limit theorem for LS estimator in simple linear EV regression models","volume":"36","author":"Miao","year":"2007","journal-title":"Theory Methods"},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"2172","DOI":"10.1080\/03610926.2011.558663","article-title":"Asymptotic normality of estimators in heteroscedastic semi-parametric model with strong mixing errors","volume":"41","author":"Zhang","year":"2012","journal-title":"Commun. Stat. Theory Methods"}],"container-title":["Symmetry"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2073-8994\/16\/11\/1544\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T16:34:46Z","timestamp":1760114086000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2073-8994\/16\/11\/1544"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2024,11,18]]},"references-count":28,"journal-issue":{"issue":"11","published-online":{"date-parts":[[2024,11]]}},"alternative-id":["sym16111544"],"URL":"https:\/\/doi.org\/10.3390\/sym16111544","relation":{},"ISSN":["2073-8994"],"issn-type":[{"type":"electronic","value":"2073-8994"}],"subject":[],"published":{"date-parts":[[2024,11,18]]}}}