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In regression analysis, not all pattern of regression curve is known due to absence of prior information about the kind of relationship between response and predictor variable. In this case, nonparametric regression becomes an alternative solution since there is no assumption about parametric form. There are several functions in nonparametric regression one of which is truncated spline that is more flexible to fit the data, good at visual interpretation, and able to handle data that have changed behavior at certain subintervals. Moreover, some application involves more than one response variables that are correlated between responses. Therefore, this study aims to obtain the curve estimation of truncated spline estimators on bi-response nonparametric regression along with estimation of error variance-covariance matrix. The curve estimation of the truncated spline estimator was obtained by Weighted Least Square (WLS) optimization with Generalized Cross Validation (GCV) as optimal knot point selection method. Then, the curve estimation of the model was applied to a real dataset of the 2019 Human Development Index (HDI) and Gender Development Index (GDI) in East Java Province, Indonesia. HDI and GDI become indicators of Sustainable Development Goals (SDGs) achievement, particularly social and economic pillars. An adequate coefficient determination from the best model indicates that the model provides good performance in modeling the data. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]