Spline smoothing for multi-response nonparametric regression model in case of heteroscedasticity of variance

Problem statement: Assume that data (y_ki, t_ki), k = 1,2,..., p; i = 1,2,...,n_k where n_k represents the number of repeated measurement of k^th object follows multi-response nonparametric regression model with variances of errors are heteroscedastic. Nonparametric regression curves are unknown and assumed to be smooth which are contained in Sobolev space. Random Errors are independent and normally distributed with zero means and unequal of variances. Approach: Smoothing spline can be used to estimate the nonparametric regression curve by carrying out the penalized weighted least-squares optimation. Therefore, reproducing kernel Hilbert space approach is applied to carry out the penalized weighted least-squares optimation. Results: In this study we consider the heteroscedastic multi-response nonparametric regression model and give a mathematical statistics method for obtaining the weighted spline estimator to estimate heteroscedastic multi-response nonparametric regression curves. Conclusion: The reproducing kernel Hilbert space approach gives solution of penalized weighted least-squares optimation for estimating heteroscedastic multi-response nonparametric regression curve which gives the weighted spline estimator. The estimator obtained is a biased estimator for nonparametric regression curve. However, the estimator is linear in observation.