The semiparametric regression curve estimation by using mixed truncated spline and fourier series model

Abstract

In simple terms, semiparametric regression is a model that combines parametric and nonparametric models. The use of two different components in semiparametric regression practically makes this model broader and developed rapidly in theoretical respect. There are several estimators where two of them are truncated spline and fourier series. Spline has the characteristic of changing patterns at certain sub-intervals while fourier series are smooth and follow the pattern repeated at certain intervals. Furthermore, in multivariable nonparametric regression, it is possible to use different estimators for each predictor. This has encouraged researcher to develop studies with mixed or combined estimators. Ordinary Least Square (OLS) as one of the most common estimation methods cannot be directly used in nonparametric regression because the shape of the curve is unknown. Hence, the OLS method is modified with conditional optimization and referred to Penalized Least Square (PLS). The semiparametric regression curve estimation obtained in this study applied to the Human Development Index (HDI) in 37 regencies across East Java. Based on data from BPS-Statistics of East Java Province, East Java's HDI is the lowest among six provinces on Java island and slightly lower than Indonesia's HDI. Therefore, further studies on East Java's HDI becomes important. In this regard, the objective of this research is to obtain an estimator of multivariable semiparametric regression curve using mixed truncated spline and fourier series model and applying the data of HDI in East Java. The method of selecting smoothing parameter using minimum Generalized Cross Validation (GCV) and the best model was obtained with two knots-two oscillation with minimum GCV equals to 4.58531 which has R2=89.20%. Model interpretations are generally divided for each predictor variable and due to R2 obtained, it can also be said that the model obtained can explain the relationship between response and predictor variables.