Estimates for the generalized cross-validation function via an extrapolation and statistical approach

Generalized cross-validation (GCV) is a popular tool for specifying the tuning parameter in linear regression model or equivalently the regularization parameter in Tikhonov regularization. In this work, we are concerned with the estimation and minimization of the GCV function by using a combination of an extrapolation procedure and a statistical approach. In particular, we derive families of estimates for the GCV function. By minimizing the estimated GCV function over a grid of values, a GCV estimate of the regularization parameter is achieved. We present several numerical examples to illustrate the effectiveness of the derived families of estimates for approximating the regularization parameter for several linear discrete ill-posed problems.