Estimation confidence interval for mixed models of spline truncated and fourier series in semiparametric regression

Regression is one of the analysis methods used to determine the pattern of the relationship between response variables and predictor variables. This relationship pattern can be approximated using three approaches, namely parametric regression, nonparametric regression, and semiparametric regression. Some commonly used nonparametric regression approaches include Kernel, Spline, Multivariate Adaptive Regression Spline (MARS), Wavelet, Fourier Series, and others. Interval estimation is an important part of inferential statistics. Confidence intervals for parameters in regression can be used to determine which predictor variables significantly affect the response variable. In this study, interval estimation is developed for model parameters using several predictor variables that have different relationship patterns. A mixed semiparametric regression approach of truncated spline and fourier series is used to explain the relationship pattern. To obtain the shortest interval estimation of the model parameters, the pivotal quantity approach is used. The estimation method used to obtain the parameter estimatior is the Maximum Likelihood Estimation (MLE) method. From the research results, the confidence interval is obtained when² is known and the confidence interval when ² is unknown. This study has successfully obtained the derivation process to obtain the estimator of a mixed semiparametric regression model consisting of a truncated spline estimator and a fourier series. The results of this theoretical study show that the Maximum Likelihood Estimation (MLE) method works simultaneously to obtain the smoothing parameter estimator and the parameters of the mixed semiparametric regression model resulting from the combination of the truncated spline and the additively separable fourier series.