Revisiting Akaike’s Final Prediction Error and the Generalized Cross Validation Criteria in Regression from the Same Perspective: From Least Squares to Ridge Regression and Smoothing Splines

In regression, despite being both aimed at estimating the Mean Squared Prediction Error (MSPE), Akaike’s Final (FPE) and Generalized Cross Validation (GCV) selection criteria are usually derived from two quite different perspectives. Here, settling on most commonly accepted definition of MSPE as expectation squared prediction error loss, we provide theoretical expressions for it, valid any linear model (LM) fitter, be it under random or non designs. Specializing these each them, able to derive closed formulas some popular LM fitters: Ordinary Least Squares (OLS), with without a full column rank design matrix; Ridge latter embedding smoothing splines fitting. For fitters, then deduce computable estimate which turns out coincide FPE. Using slight variation, similarly get class estimates coinciding classical GCV formula those same fitters.