The Curve Estimation Nonparametric Regression Multiresponse Mixed with Truncated Spline, Fourier Series, and Kernel

This study formulates a nonparametric regression model for multiresponse data by combining three estimators: truncated spline, Fourier series, and kernel function. Each estimator captures specific characteristics. Truncated spline capture local traits with knot points, while fourier series capture periodic patterns and kernel estimators provide flexible smoothing for unknown functional forms. The model proposed is under an additive assumption where each predictor contributes independently to each response. Estimation is done with Weighted Least Squares (WLS) method which is efficient in managing the correlations between the multiresponse variables. The final multiresponse nonparametric regression curve estimator combining truncated spline, Fourier series, and kernel is given by \hat{\mu} = \hat{f} + \hat{g} + \hat{h}\] obtained by solving the WLS optimization problem: [\min_{\boldsymbol{\beta}, \boldsymbol{\alpha}} \{ \boldsymbol{\varepsilon}' W \boldsymbol{\varepsilon} \} =\min_{\boldsymbol{\beta}, \boldsymbol{\alpha}} \left\{ (\mathbf{y}^* - U \boldsymbol{\beta} - Z \boldsymbol{\alpha})' W (\mathbf{y}^* - U \boldsymbol{\beta} - Z \boldsymbol{\alpha}) \right\}.\]. The solution to this problem results in the mixed estimator, which can be expressed as: \[\hat{\boldsymbol{\mu}} = E \mathbf{y} \quad \text{with} \quad E = UB + ZA + T.\]