Selection of dimension and basis for density estimation and selection of dimension, basis and error distribution for regression

When approximations of the form are used in regression or density estimation, the dimension m controls the smoothness and goodness of fit of the approximation. For this type of approximation, Akaike's Information Criterion (AIC) provides a balance between smoothness and goodness of fit, extending maximum likelihood methods from estimation of parameters for a specified dimension (model) to the selection of dimension for a given basis (ψ i (x)'s). Some basis will give a smaller bias for a given dimension than others and also may suggest a parametric model for a given data. In this paper, use of AIC is first extended from selection of dimension for a given basis to selection of basis for density estimation and regression. Next, it is extended to model selection (basis and dimension) under different error distributions leading to robust model selection for regression.