Mixture Spline Smoothing and Kernel Estimator in Multi-Response Nonparametric Regression

In previous research about multi-response nonparametric regression models, each predictor variable is considered to have the same pattern concerning each response variable. In contrast, multi-response cases are often encountered with different patterns among the predictor variables. Therefore, a mixture estimator in multi-response nonparametric regression needs to be developed. This study proposes an additive mixture of Spline Smoothing and Kernel estimator in multi-response nonparametric regression. Our approach can handle the previously mentioned issue in a multi-response nonparametric regression problem, i.e., some predictors showing changing patterns in certain sub-intervals, such as Spline Smoothing patterns, and other predictors exhibiting random patterns, commonly modeled using Kernel regression. A two-stage estimation procedure, i.e., Penalized Weighted Least Square followed by Weighted Least Square, was used to obtain this mixture estimator. Furthermore, a simulation study and real data analysis were conducted to illustrate the performance of the proposed multi-response mixture estimator. The results indicate that the proposed multi-response mixture estimator can be applied appropriately and gives satisfactory results with a coefficient of determination (R²) close to 1 and a Mean Absolute Percentage Error (MAPE) of less than 5%.