This paper studies, under the setting of spline regression, the connection between finite-sample properties of selection criteria and their asymptotic counterparts, focusing on bridging the gap between the two. We introduce a bias-variance decomposition of the prediction error, using which it is shown that in the asymptotics the bias term dominates the variability term, providing an explanation of the gap. A geometric exposition is provided for intuitive understanding. The theoretical and geometric results are illustrated through a numerical example.
"From finite sample to asymptotics: A geometric bridge for selection criteria in spline regression." Ann. Statist. 32 (6) 2444 - 2468, December 2004. https://doi.org/10.1214/009053604000000841