Properties of The Mixed Smoothing Spline and Fourier Series Estimators in Nonparametric Regression

In regression analysis, the pattern of the relationship between two or more variables is not always a parametric pattern such as linear, quadratic, cubic and others. There are many cases where the relationship pattern between variables is nonparametric pattern. In parametric regression the shape of the regression curve is assumed to be known. In contrast to the parametric approach, in nonparametric regression the shape of the regression curve is assumed to be unknown. The regression curve is only assumed to be smooth in the sense that it is contained in a certain function space. Researchers mostly develop one type of estimator in nonparametric regression. However, in reality, data with mixed patterns are often encountered, especially data patterns that partly change at certain sub-intervals and partly follow a pattern that repeats itself in a certain trend. In dealing with the mixed pattern, this paper will explain the combination of the Smoothing Spline function and the Fourier Series. Theoretical research is focused on the estimator model and its properties. The estimator model is solved by minimizing the Penalized Least Square (PLS). The mixed estimator properties of Smoothing Spline and Fourier Series in multivariable nonparametric regression are linear classes and are biased in small samples.