{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,3,6]],"date-time":"2026-03-06T17:58:28Z","timestamp":1772819908078,"version":"3.50.1"},"reference-count":50,"publisher":"MDPI AG","issue":"4","license":[{"start":{"date-parts":[[2024,3,25]],"date-time":"2024-03-25T00:00:00Z","timestamp":1711324800000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/501100008463","name":"Airlangga Research Fund, Universitas Airlangga, Indonesia","doi-asserted-by":"publisher","award":["312\/UN3.15\/PT\/2023"],"award-info":[{"award-number":["312\/UN3.15\/PT\/2023"]}],"id":[{"id":"10.13039\/501100008463","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>In data analysis using a nonparametric regression approach, we are often faced with the problem of analyzing a set of data that has mixed patterns, namely, some of the data have a certain pattern and the rest of the data have a different pattern. To handle this kind of datum, we propose the use of a mixed estimator. In this study, we theoretically discuss a developed estimation method for a nonparametric regression model with two or more response variables and predictor variables, and there is a correlation between the response variables using a mixed estimator. The model is called the multiresponse multipredictor nonparametric regression (MMNR) model. The mixed estimator used for estimating the MMNR model is a mixed estimator of smoothing spline and Fourier series that is suitable for analyzing data with patterns that partly change at certain subintervals, and some others that follow a recurring pattern in a certain trend. Since in the MMNR model there is a correlation between responses, a symmetric weight matrix is involved in the estimation process of the MMNR model. To estimate the MMNR model, we apply the reproducing kernel Hilbert space (RKHS) method to penalized weighted least square (PWLS) optimization for estimating the regression function of the MMNR model, which consists of a smoothing spline component and a Fourier series component. A simulation study to show the performance of proposed method is also given. The obtained results are estimations of the smoothing spline component, Fourier series component, MMNR model, weight matrix, and consistency of estimated regression function. In conclusion, the estimation of the MMNR model using the mixed estimator is a combination of smoothing spline component and Fourier series component estimators. It depends on smoothing and oscillation parameters, and it has linear in observation and consistent properties.<\/jats:p>","DOI":"10.3390\/sym16040386","type":"journal-article","created":{"date-parts":[[2024,3,25]],"date-time":"2024-03-25T12:47:11Z","timestamp":1711370831000},"page":"386","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":10,"title":["Estimation of Multiresponse Multipredictor Nonparametric Regression Model Using Mixed Estimator"],"prefix":"10.3390","volume":"16","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 Sciences, Faculty of Science and Technology, Airlangga University, Surabaya 60115, Indonesia"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Budi","family":"Lestari","sequence":"additional","affiliation":[{"name":"Research Group of Statistical Modeling in Life Sciences, 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"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-6572-4083","authenticated-orcid":false,"given":"I Nyoman","family":"Budiantara","sequence":"additional","affiliation":[{"name":"Research Group of Statistical Modeling in Life Sciences, Faculty of Science and Technology, Airlangga University, Surabaya 60115, Indonesia"},{"name":"Department of Statistics, Faculty of Sciences and Data Analytics, Sepuluh Nopember Institute of Technology, Surabaya 60111, Indonesia"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"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"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2024,3,25]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","unstructured":"Eubank, R.L. 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