{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,3,7]],"date-time":"2026-03-07T00:17:30Z","timestamp":1772842650389,"version":"3.50.1"},"reference-count":45,"publisher":"MDPI AG","issue":"2","license":[{"start":{"date-parts":[[2022,2,6]],"date-time":"2022-02-06T00:00:00Z","timestamp":1644105600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/501100008463","name":"Airlangga University","doi-asserted-by":"publisher","award":["373\/UN3.14\/PT\/2020"],"award-info":[{"award-number":["373\/UN3.14\/PT\/2020"]}],"id":[{"id":"10.13039\/501100008463","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>A multiresponse multipredictor semiparametric regression (MMSR) model is a combination of parametric and nonparametric regressions models with more than one predictor and response variables where there is correlation between responses. Due to this correlation we need to construct a symmetric weight matrix. This is one of the things that distinguishes it from the classical method, which uses a parametric regression approach. In this study, we theoretically developed a method of determining a confidence interval for parameters in a MMSR model based on a truncated spline, and investigating asymptotic properties of estimator for parameters in a MMSR model, especially consistency and asymptotic normality. The weighted least squares method was used to estimate the MMSR model. Next, we applied a pivotal quantity method, a Cramer\u2013Wold theorem, and a Slutsky theorem to determine the confidence interval, investigate consistency, and asymptotic normality properties of estimator for parameters in a MMSR model. The obtained results were that the estimated regression function is linear to observation. We also obtained a 1001\u2212\u03b1% confidence interval for parameters in the MMSR model, and the estimator for parameters in MMSR model was consistent and asymptotically normally distributed. In the future, these obtained results can be used as a theoretical basis in designing a standard toddlers growth chart to assess nutritional status.<\/jats:p>","DOI":"10.3390\/sym14020336","type":"journal-article","created":{"date-parts":[[2022,2,6]],"date-time":"2022-02-06T20:40:18Z","timestamp":1644180018000},"page":"336","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":13,"title":["Consistency and Asymptotic Normality of Estimator for Parameters in Multiresponse Multipredictor Semiparametric Regression Model"],"prefix":"10.3390","volume":"14","author":[{"ORCID":"https:\/\/orcid.org\/0000-0003-1592-4671","authenticated-orcid":false,"given":"Nur","family":"Chamidah","sequence":"first","affiliation":[{"name":"Department of Mathematics, Faculty of Science and Technology, Airlangga University, Surabaya 60115, Indonesia"},{"name":"Research Group of Statistical Modeling in Life Science, Faculty of Science and Technology, Airlangga University, Surabaya 60115, Indonesia"}]},{"given":"Budi","family":"Lestari","sequence":"additional","affiliation":[{"name":"Research Group of Statistical Modeling in Life Science, Faculty of Science and Technology, Airlangga University, Surabaya 60115, Indonesia"},{"name":"Department of Mathematics, Faculty of Mathematics and Natural Sciences, The University of Jember, Jember 68121, Indonesia"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-6572-4083","authenticated-orcid":false,"given":"I. Nyoman","family":"Budiantara","sequence":"additional","affiliation":[{"name":"Department of Statistics, Faculty of Sciences and Data Analytics, Sepuluh Nopember Institute of Technology, Surabaya 60111, Indonesia"}]},{"given":"Toha","family":"Saifudin","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Faculty of Science and Technology, Airlangga University, Surabaya 60115, Indonesia"},{"name":"Research Group of Statistical Modeling in Life Science, Faculty of Science and Technology, Airlangga University, Surabaya 60115, Indonesia"}]},{"given":"Riries","family":"Rulaningtyas","sequence":"additional","affiliation":[{"name":"Research Group of Statistical Modeling in Life Science, Faculty of Science and Technology, Airlangga University, Surabaya 60115, Indonesia"},{"name":"Department of Physics, Faculty of Science and Technology, Airlangga University, Surabaya 60115, Indonesia"}]},{"given":"Aryati","family":"Aryati","sequence":"additional","affiliation":[{"name":"Research Group of Statistical Modeling in Life Science, Faculty of Science and Technology, Airlangga University, Surabaya 60115, Indonesia"},{"name":"Department of Clinical Pathology, Faculty of Medicine, Airlangga University, Surabaya 60131, Indonesia"}]},{"given":"Puspa","family":"Wardani","sequence":"additional","affiliation":[{"name":"Research Group of Statistical Modeling in Life Science, Faculty of Science and Technology, Airlangga University, Surabaya 60115, Indonesia"},{"name":"Department of Clinical Pathology, Faculty of Medicine, Airlangga University, Surabaya 60131, Indonesia"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-8393-1270","authenticated-orcid":false,"given":"Dursun","family":"Aydin","sequence":"additional","affiliation":[{"name":"Department of Statistics, Faculty of Science, Mu\u011fla S\u0131tk\u0131 Ko\u00e7man University, Mu\u011fla 48000, Turkey"}]}],"member":"1968","published-online":{"date-parts":[[2022,2,6]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"012067","DOI":"10.1088\/1742-6596\/1397\/1\/012067","article-title":"Modeling of hypertension risk factors using local linear of additive nonparametric logistic regression","volume":"1397","author":"Ana","year":"2019","journal-title":"J. Phys. Conf. Ser."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"460","DOI":"10.2991\/jsta.d.201016.001","article-title":"Local linear regression estimator on the boundary correction in nonparametric regression estimation","volume":"19","author":"Cheruiyot","year":"2020","journal-title":"J. Statist. Theory Appl."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"2109","DOI":"10.11591\/eei.v9i5.2021","article-title":"Identification the number of mycobacterium tuberculosis based on sputum image using local linear estimator","volume":"9","author":"Chamidah","year":"2020","journal-title":"Bullet. Elect. Eng. Inform. (BEEI)"},{"key":"ref_4","first-page":"2749","article-title":"Bias reduction for nonparametric and semiparametric regression models","volume":"28","author":"Cheng","year":"2018","journal-title":"Statistica Sinica"},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"348","DOI":"10.1198\/jasa.2009.0114","article-title":"A design-adaptive local polynomial estimator for the errors-in-variables problem","volume":"104","author":"Delaigle","year":"2009","journal-title":"J. Amer. Stat. Assoc."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"1271","DOI":"10.1081\/STA-100104745","article-title":"Local polynomial regression estimation with correlated errors","volume":"30","year":"2001","journal-title":"Comm. Statist. Theory Methods"},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"881","DOI":"10.1051\/ps\/2014009","article-title":"Local polynomial estimation of the mean function and its derivatives based on functional data and regular designs","volume":"18","author":"Benhenni","year":"2014","journal-title":"ESAIM Probab. Stat."},{"key":"ref_8","first-page":"58","article-title":"On local polynomial regression estimators in finite populations","volume":"5","author":"Kikechi","year":"2020","journal-title":"Int. J. Stats. Appl. Math."},{"key":"ref_9","doi-asserted-by":"crossref","unstructured":"Wand, M.P., and Jones, M.C. (1995). Kernel Smoothing, Chapman and Hall\/CRC. [1st ed.].","DOI":"10.1007\/978-1-4899-4493-1"},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"179","DOI":"10.4236\/ojs.2013.33020","article-title":"Strong consistency of kernel regression estimate","volume":"3","author":"Cui","year":"2013","journal-title":"Open J. Stats."},{"key":"ref_11","first-page":"1955","article-title":"Kernel regression in the presence of correlated errors","volume":"12","author":"Suykens","year":"2011","journal-title":"J. Mach. Learn. Res."},{"key":"ref_12","doi-asserted-by":"crossref","unstructured":"Wahba, G. (1990). Spline Models for Observational Data, SIAM.","DOI":"10.1137\/1.9781611970128"},{"key":"ref_13","doi-asserted-by":"crossref","unstructured":"Eubank, R.L. (1999). Nonparametric Regression and Spline Smoothing, Marcel Dekker. [2nd ed.].","DOI":"10.1201\/9781482273144"},{"key":"ref_14","doi-asserted-by":"crossref","unstructured":"Wang, Y. (2011). Smoothing Splines: Methods and Applications, Taylor & Francis Group.","DOI":"10.1201\/b10954"},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"2282","DOI":"10.1016\/j.jmva.2010.06.001","article-title":"M-type smoothing spline ANOVA for correlated data","volume":"101","author":"Liu","year":"2010","journal-title":"J. Multivar. Anal."},{"key":"ref_16","first-page":"1","article-title":"Estimating mean arterial pressure affected by stress scores using spline nonparametric regression model approach","volume":"2020","author":"Chamidah","year":"2020","journal-title":"Commun. Math. Biol. Neurosci."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"86","DOI":"10.1214\/ss\/1038425655","article-title":"Flexible smoothing with B-splines and penalties","volume":"11","author":"Eilers","year":"1996","journal-title":"Statist. Sci."},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"3","DOI":"10.1177\/0962280218820406","article-title":"Efficient estimation of a linear transformation model for current status data via penalized splines","volume":"29","author":"Lu","year":"2020","journal-title":"Stat. Meth. Medic. Res."},{"key":"ref_19","first-page":"377","article-title":"Spline smoothing for bivariate data with applications to association between hormones","volume":"10","author":"Wang","year":"2000","journal-title":"Stat. Sinica"},{"key":"ref_20","doi-asserted-by":"crossref","unstructured":"Yilmaz, E., Ahmed, S.E., and Aydin, D. (2020). A-Spline regression for fitting a nonparametric regression function with censored data. Stats, 3.","DOI":"10.3390\/stats3020011"},{"key":"ref_21","first-page":"253","article-title":"A comparison of the nonparametric regression models using smoothing spline and kernel regression","volume":"36","author":"Aydin","year":"2007","journal-title":"World Acad. Sci. Eng. Tech."},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"012064","DOI":"10.1088\/1742-6596\/1397\/1\/012064","article-title":"Smoothing parameter selection method for multiresponse nonparametric regression model using spline and kernel estimators approaches","volume":"1397","author":"Lestari","year":"2019","journal-title":"J. Phy. Conf. Ser."},{"key":"ref_23","first-page":"47","article-title":"Choice of bandwidth for nonparametric regression models using kernel smoothing: A simulation study","volume":"26","author":"Aydin","year":"2016","journal-title":"Int. J. Sci. Basic Appl. Research (IJSBAR)"},{"key":"ref_24","first-page":"90","article-title":"Kernel and regression spline smoothing techniques to estimate coefficient in rates model and its application in psoriasis","volume":"33","author":"Osmani","year":"2019","journal-title":"Medic. J. Islamic Repub. Iran (MJIRI)"},{"key":"ref_25","first-page":"1177","article-title":"Comparison of smoothing and truncated spline estimators in estimating blood pressures models","volume":"5","author":"Fatmawati","year":"2019","journal-title":"Int. J. Innov. Creat. Change (IJICC)"},{"key":"ref_26","first-page":"533","article-title":"Spline estimator and its asymptotic properties in multiresponse nonparametric regression model","volume":"42","author":"Lestari","year":"2020","journal-title":"Songklanakarin J. Sci. Tech. (SJST)"},{"key":"ref_27","doi-asserted-by":"crossref","unstructured":"Mariati, M.P.A.M., Budiantara, I.N., and Ratnasari, V. (2021). The application of mixed smoothing spline and Fourier series model in nonparametric regression. Symmetry, 13.","DOI":"10.3390\/sym13112094"},{"key":"ref_28","doi-asserted-by":"crossref","unstructured":"Ruppert, D., Wand, M.P., and Carroll, R.J. (2003). Semiparametric Regression, Cambridge University Press.","DOI":"10.1017\/CBO9780511755453"},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"244","DOI":"10.1111\/j.2517-6161.1986.tb01407.x","article-title":"Spline smoothing in a partly linear model","volume":"48","author":"Heckman","year":"1986","journal-title":"J. R. Stats. Soc. Ser. B."},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"1","DOI":"10.31642\/JoKMC\/2018\/020301","article-title":"Spline semiparametric regression models","volume":"2","author":"Mohaisen","year":"2015","journal-title":"J. Kufa Math. Comp."},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"724","DOI":"10.1360\/01ys0206","article-title":"Iterative weighted partial spline least squares estimation in semiparametric modeling of longitudinal data","volume":"46","author":"Sun","year":"2003","journal-title":"Science in China Series A (Mathematics)"},{"key":"ref_32","first-page":"45","article-title":"Designing local standard growth charts of children in East Java province using a local linear estimator","volume":"13","author":"Chamidah","year":"2020","journal-title":"Int. J. Innov. Creat. Change (IJICC)"},{"key":"ref_33","first-page":"70","article-title":"Comparison of regression models based on nonparametric estimation techniques: Prediction of GDP in Turkey","volume":"1","author":"Aydin","year":"2007","journal-title":"Int. J. Math. Models Methods Appl. Sci."},{"key":"ref_34","doi-asserted-by":"crossref","first-page":"985","DOI":"10.1080\/00949655.2019.1572757","article-title":"Estimation of semiparametric regression model with right-censored high-dimensional data","volume":"89","author":"Ahmed","year":"2019","journal-title":"J. Stat. Comp. Simul."},{"key":"ref_35","first-page":"1155","article-title":"M-Type smoothing splines in nonparametric and semiparametric regression models","volume":"7","author":"Gao","year":"1997","journal-title":"Stat. Sinica"},{"key":"ref_36","doi-asserted-by":"crossref","first-page":"165","DOI":"10.1198\/jcgs.2009.0010","article-title":"Smoothing spline semiparametric nonlinear regression models","volume":"18","author":"Wang","year":"2009","journal-title":"J. Comp. Graph. Stats."},{"key":"ref_37","doi-asserted-by":"crossref","first-page":"161","DOI":"10.3844\/jmssp.2013.161.168","article-title":"Smoothing spline in semiparametric additive regression model with Bayesian approach","volume":"9","author":"Diana","year":"2013","journal-title":"J. Math. Stats."},{"key":"ref_38","doi-asserted-by":"crossref","first-page":"723","DOI":"10.1016\/j.jmva.2010.11.001","article-title":"Empirical likelihood for semiparametric regression model with missing response data","volume":"102","author":"Xue","year":"2011","journal-title":"J. Multivar. Anal."},{"key":"ref_39","doi-asserted-by":"crossref","first-page":"489","DOI":"10.3844\/jmssp.2012.489.499","article-title":"On multiresponse semiparametric regression model","volume":"8","author":"Wibowo","year":"2012","journal-title":"J. Math. Stats."},{"key":"ref_40","first-page":"637","article-title":"Simultaneous confidence intervals for semiparametric logistics regression and confidence regions for the multi-dimensional effective dose","volume":"20","author":"Li","year":"2010","journal-title":"Stat. Sinica"},{"key":"ref_41","first-page":"1","article-title":"Estimating regression function of multiresponse semiparametric regression model using smoothing spline","volume":"55","author":"Lestari","year":"2020","journal-title":"J. Southwest Jiaotong Univ."},{"key":"ref_42","first-page":"14","article-title":"Confidence interval of multiresponse semiparametric regression model parameters using truncated spline","volume":"4","author":"Hidayati","year":"2020","journal-title":"Int. J. Acad. Appl. Res. (IJAAR)"},{"key":"ref_43","unstructured":"Sahoo, P. (2013). Probability and Mathematical Statistics, University of Louisville."},{"key":"ref_44","first-page":"290","article-title":"Some theorems on distribution functions","volume":"11","author":"Wold","year":"1936","journal-title":"J. London Math. Soc."},{"key":"ref_45","doi-asserted-by":"crossref","unstructured":"Sen, P.K., and Singer, J.M. (1993). Large Sample in Statistics: An Introduction with Applications, Chapman & Hall.","DOI":"10.1007\/978-1-4899-4491-7"}],"container-title":["Symmetry"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2073-8994\/14\/2\/336\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T22:15:04Z","timestamp":1760134504000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2073-8994\/14\/2\/336"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2022,2,6]]},"references-count":45,"journal-issue":{"issue":"2","published-online":{"date-parts":[[2022,2]]}},"alternative-id":["sym14020336"],"URL":"https:\/\/doi.org\/10.3390\/sym14020336","relation":{},"ISSN":["2073-8994"],"issn-type":[{"value":"2073-8994","type":"electronic"}],"subject":[],"published":{"date-parts":[[2022,2,6]]}}}