Multivariable Semiparametric Regression Model with Combined Estimator of Fourier Series and Kernel

We purpose a combined estimator of Fourier series and Kernel on semiparametric regression. This method is used to resolve the problem of regression modeling, when the relationship between the response variable and the predictor variables most follow a certain pattern, partly have a repetitive pattern, and some others not follow a specific pattern. Moreover, this method depends on oscillation, smoothing parameter and bandwidth. The purpose of this research are to obtained the estimator of semiparametric regression model with combined estimator of Fourier series and Kernel using Penalized Least Square method (PLS). The result show that the PLS estimation produces the estimator of parametrik linier regression, the estimator of Fourier series, the estimator of Kernel, and also the combined estimator of Fourier series and Kernel in semiparametric regression model.


Introduction
Semiparametric regression is a statistical methods used to estimate the relationship pattern of the predictor variable and response variable, when a case in regression analysis contains two components; parametric component and nonparametric component. Based on several previous research that has been conducted by researchers, mostly they use the same estimator approach for all or some of the predictor variables. Meanwhile, in many cases, the data pattern of each predictor variables are not always identical. Therefore, to solve these problems, we need a more proper estimator that can be used to approximate the data pattern. In semiparametric regression, there are many functions that can be used to approximate the data pattern, especially Fouries series and Kernel.
According to [1], estimation of Fourier series is capable of handling data that is smooth character and follow the pattern repeated at certain interval, previous research have been investigate by researchers as Amato [2], Asrini [3], Bilodeau [1], Pane [4], and Sudiarsa [5]. The kernel is one of the frequenlty used estimators in semiparametric and nonparametric regression. This estimator has more flexible shape, simple mathematics calculation, and achieves convergence more rapidly. Several researches on Kernel estimator have been conducted by some researchers such as Nadaraya [6], Budiantara et al [7], and Speackman [8].
Therefore, this research focus on semiparametric regression model with combined estimator between Fourier series and Kernel obtained through PLS method.

Semiparametric Regression Model
Given the data ( ) Linear function with p predictor variable can be written as : ( )

parameters of the linier function
Fourier series function with one predictor variable can be written as : Hence, a semiparametric regression model in equation (1) can be written as follows : y X Ga Dy

Multivariable Semiparametric Regression with Combined Estimator between Fourier Series and Kernel
The combination estimator of Fourier series and Kernel in semiparametric regression, ( ) Several lemmas are needed to complete this PLS optimization.

Lemma 1
If the function of Fourier series is ( ) g t as in equation (3), then the penalty is   T T Q a n Y X Ga Dy Y X Ga Dy a Ua  T  T  T  T  T  T  T  T   T  T  T  T  T n X I D y n X X n X Ga n a G I D y n a G X n a G Ga a Ua

y I D I D y n X I D y n a G I D y
The derivation of ( , ) Q a β to β and a as follows : Equalizing the first derivatif of sum square error with zero, for β and â are obtained : Then substitution β into â , so : ˆB y a = where : The estimator of semiparametric with combined Fourier series and Kernel, obtained :

Conclusion
The results shows that the estimator of combination between Fourier series and Kernel function in semiparametric regression was obtined through the PLS optimization, as follows : The estimator of combination between Fourier series and Kernel function is