Approximation of Least Squares Regression on Nested Subspaces

For a regression model yᵢ = (xᵢ) + ᵢ, the unknown function is estimated by least squares on a subspace ₘ = span\₁, , , ₘ\, where the basis functions ᵢ are predetermined and m is varied. Assuming that the design is suitably approximated by an asymptotic design measure, a general method is presented for approximating the bias and variance in a scale of Hilbertian norms natural to the problem. The general theory is illustrated with two examples: truncated Fourier series regression and polynomial regression. For these examples, we give rates of convergence of derivative estimates in (weighted) L₂ norms and establish consistency in supremum norm.