Smoothing splines is a well stablished method in non-parametric statistics, although the selection of the smoothness degree of the regression function is rarely addressed and, instead, a two times differentiable function, i.e. cubic smoothing spline, is assumed. For a general regression function there is no known method that can identify the smoothness degree under the presence of correlated errors. This apparent disregard in the literature can be justified because the condition number of the solution increases with the smoothness degree of the function, turning the estimation unstable. In this thesis we introduce an exact expression for the Demmler-Reinsch basis constructed as the solution of an ordinary differential equation, so that the estimation can be carried out for an arbitrary smoothness degree, and under the presence of correlated errors, without affecting the condition number of the solution. We provide asymptotic properties of the proposed estimators and conduct simulation experiments to study their finite sample properties. We expect this new approach to have a direct impact on related methods that use smoothing splines as a building block. In this direction, we present extensions of the method to signal extraction and functional principal component analysis. The empirical relevance to our findings in these areas of statistics is shown in applications for agricultural economics and biophysics. R packages of the implementation of the developed methods are also provided.
