Research Article Nonparametric Regression Model for Longitudinal Data with Mixed Truncated Spline and Fourier Series Made Ayu Dwi Octavanny , I. Nyoman Budiantara , Heri Kuswanto , and Dyah Putri Rahmawati Department of Statistics, Faculty of Science and Data Analytics, Institut Teknologi Sepuluh Nopember, Surabaya 60111, Indonesia Correspondence should be addressed to I. Nyoman Budiantara; nyomanbudiantara65@gmail.com Received 30 July 2020; Revised 24 November 2020; Accepted 1 December 2020; Published 10 December 2020 Academic Editor: Victor Kovtunenko Copyright © 2020 Made Ayu Dwi Octavanny et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Existing literature in nonparametric regression has established a model that only applies one estimator to all predictors. This study is aimed at developing a mixed truncated spline and Fourier series model in nonparametric regression for longitudinal data. The mixed estimator is obtained by solving the two-stage estimation, consisting of a penalized weighted least square (PWLS) and weighted least square (WLS) optimization. To demonstrate the performance of the proposed method, simulation and real data are provided. The results of the simulated data and case study show a consistent finding. 1. Introduction Regression analysis is aimed at modeling the association between the predictor and the response. If the data pattern shows an unknown regression curve, nonparametric regres- sion is used [1]. However, if the form of the regression curve is known, parametric regression can be applied [2]. Addition- ally, nonparametric regression has high flexibility because the data is expected to find its regression curve estimation form without being influenced by the researcher’s subjectivity [3]. In this study, we have analyzed several models such as kernel, spline [4–7], and Fourier series [8]. A spline estimator, which has an excellent ability to han- dle data with changes at subspecified intervals [9], was obtained using penalized least square optimization [10] and the Bayesian approach [11]. A spline estimator can be applied for cross-sectional data as well as longitudinal data. Additionally, several studies on nonparametric regression for longitudinal data have been addressed using kernel esti- mator [12, 13], generalized spline regression [14], and mixed-effects model [7]. Fourier series, which is useful to explain curves that show sine and cosine waves, is generally used if the data pattern is unknown and there is a tendency to iterate. A considerable amount of research has used only one estimator for each predictor. However, because each predic- tor can have a different pattern, it was proposed to develop a mixed estimator. Recently, Sudiarsa et al. [15] discussed a study of the mixed estimator of the truncated spline and Fou- rier series. The study, which only discussed cross-sectional data, did not obtain a model for each subject as it did not include longitudinal data. Consequently, this study cannot be used to investigate response behavior based on the time change. Although some research has been carried out on a mixed estimator, no studies have explored multisubject data so far. This paper proposes a new methodology for a mixed estima- tor of the truncated spline and Fourier series in the nonpara- metric regression for longitudinal data. This study addresses the gap in previous research by obtaining a mixed estimator of the truncated spline and Fourier series in the nonparamet- ric regression for longitudinal data and applying it to simu- lated data and a case study. This study is organized as follows. We briefly explain the materials and methods used in our study in Section 2. Section 3 consists of three subsections: the developed theory, simula- tion study, and case study. We present the developed non- parametric regression theory for longitudinal data with a Hindawi Abstract and Applied Analysis Volume 2020, Article ID 4710745, 11 pages https://doi.org/10.1155/2020/4710745