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.

Pendidikan karakter harus dimulai sejak usia dini agar di masa depan dapat tercipta sumber daya kuat yang berkarakter. Tujuan dari penelitian ini adalah mendeskripsikan cara guru menanamkan karakter, kendala guru dalam menanamkan karakter, dan cara mengatasi kendala dalam menanamkan karakter pada peserta didik. Penelitian ini menggunakan metode kualitatif dengan pendekatan deskriptif. Teknik yang digunakan yaitu teknik observasi dan wawancara. Subjek dalam penelitian ini yaitu wali kelas I-VI dan Kepala Sekolah SD Negeri Sepandan. Data dianalisis secara deskriptif melalui tiga tindakan yaitu kodifikasi data, penyajian data, dan penarikan kesimpulan. Hasil dari penelitian ini adalah (1) cara guru dalam menanamkan karakter yaitu dengan cara pembiasaan, keteladanan, kantin serta koperasi kejujuran (2) kendala yang dialami yaitu faktor dari lingkungan sekitar peserta didik dan keluarga, (3) cara guru dalam mengatasi kendala penanaman karakter dengan pemberian nasihat dan home visit. Penanaman karakter akan memberikan dampak positif dalam kegiatan belajar mengajar, peserta didik menjadi lebih taan terhadap peraturan, disiplin berangkat sekolah, dan kreatif dalam pembelajaran

Nonparametric regression aims to determine the relationship between response and predictor when the data does not follow a specific pattern. Fourier series is a nonparametric regression approach which has the flexibility to follow the characteristics of data. The purpose of this study is to obtain the estimator of the nonparametric regression using Fourier series and apply the model to the fertility data. The fertility rate represented by children ever born is one of the demographic factors that determine the decline in population growth rate. The data were obtained from the Indonesia Demographic and Health Survey 2017 with children ever born as a response. The predictors are the proportion of women graduating from junior high school, the proportion of women having sex before the age of 18, the proportion of women using a modern contraceptive method, and the infant mortality rate. The relationship between response and predictors tends to have a repetitive pattern with a certain trend. The best nonparametric regression model of children ever born in Indonesia is obtained by using 3 oscillation parameters for each predictor variable with GCV = 0.0534 and R-square = 80.04%.

Mixed estimators in nonparametric regression have been developed in models with one response. The biresponse cases with different patterns among predictor variables that tend to be mixed estimators are often encountered. Therefore, in this article, we propose a biresponse nonparametric regression model with mixed spline smoothing and kernel estimators. This mixed estimator is suitable for modeling biresponse data with several patterns (response vs. predictors) that tend to change at certain subintervals such as the spline smoothing pattern, and other patterns that tend to be random are commonly modeled using kernel regression. The mixed estimator is obtained through two-stage estimation, i.e., penalized weighted least square (PWLS) and weighted least square (WLS). Furthermore, the proposed biresponse modeling with mixed estimators is validated using simulation data. This estimator is also applied to the percentage of the poor population and human development index data. The results show that the proposed model can be appropriately implemented and gives satisfactory results.

We introduce a new method for estimating the nonparametric regression curve for longitudinal data. This method combines two estimators: truncated spline and Fourier series. This estimation is completed by minimizing the penalized weighted least squares and weighted least squares. This paper also provides the properties of the new mixed estimator, which are biased and linear in the observations. The best model is selected using the smallest value of generalized cross-validation. The performance of the new method is demonstrated by a simulation study with a variety of time points. Then, the proposed approach is applied to a stroke patient dataset. The results show that simulated data and real data yield consistent findings.