Spline estimator in homoscedastic multiresponse nonparametric regression model in case of unbalanced number of observations

Estimating the multi-response nonparametric regression functions which draw association of two or more dependent variables observed Keywords and phrases covariance matrix, multi-response nonparametric regression, penalized weighted least-square, reproducing kernel Hilbert space, spline estimator. at several values of the independent variables is the main problem in the multi-response nonparametric regression analysis. Multi-response nonparametric regression functions are unknown and assumed to be smooth which are contained in Sobolev space. Spline estimator can be used to estimate the nonparametric regression function by carrying out the penalized weighted least-squares optimization. In this paper, we consider homoscedastic multi-response nonparametric regression model in case of unbalanced number of observations, and give a mathematical statistics method for obtaining the spline estimator. The results show that the reproducing kernel Hilbert space approach gives solution of penalized weighted least-squares optimization for estimating homoscedastic multi-response nonparametric regression function providing the weighted spline estimator.