EVALUATION OF THE FLEXIBILITY OF NADARAYA-WATSON KERNEL AND PENALIZED SPLINE ESTIMATORS IN BIVARIATE RESPONSE NONPARAMETRIC REGRESSION MODELS

Abstract

Nonparametric regression is a flexible approach used when the functional relationship between predictors and responses is unknown. In the context of multiple responses, bivariate nonparametric regression allows modeling two correlated response variables, such as stunting and wasting prevalence, which remain critical issues in public health. This study aims to evaluate the flexibility and performance of two nonparametric estimators, the Nadaraya-Watson Kernel and the Penalized Spline, for modeling bivariate response data. The research was conducted in two stages: (1) simulation using variations in sample sizes (50, 100, 150, 200) and error variances based on exponential and trigonometric functions, and (2) application to real data on stunting and wasting prevalence in Indonesia (2024) obtained from Statistics Indonesia (BPS), with socioeconomic and health-related predictors. Model performance was assessed using RMSE, MSE, and R-squared, complemented by MANOVA, orthogonal polynomial contrasts, and Tukey’s post-hoc test to examine significant differences across scenarios. Simulation results indicate that the Nadaraya-Watson Kernel estimator consistently outperformed the Penalized Spline, providing lower RMSE and MSE values and greater stability, particularly for larger sample sizes and smaller error variances. Orthogonal polynomial analysis revealed a quadratic relationship between sample size and RMSE, with occasional cubic patterns, while error variance consistently exhibited a quadratic trend. In the applied study, the Nadaraya-Watson Kernel with a Gaussian kernel achieved high accuracy, with an MSE of 0.00086 and an R-squared value indicating a strong model fit. However, this high R-squared value may reflect potential overfitting, which warrants further validation through cross-validation. These findings demonstrate that the Nadaraya-Watson Kernel offers an effective approach for bivariate nonparametric regression, supporting data-driven policy decisions in nutrition and public health.