Strong robust generalized cross-validation for choosing the regularization parameter

Let f λ be the Tikhonov regularized solution of a linear inverse or smoothing problem with discrete noisy data y i , i = 1, ..., n . To choose λ we propose a new strong robust GCV method denoted by R 1 GCV that is part of a family of such methods, including the basic RGCV method. R 1 GCV chooses λ to be the minimizer of γ V (λ) + (1 − γ) F 1 (λ), where V (λ) is the GCV function, F 1 (λ) is a certain approximate total measure of the influence of each data point on f λ with respect to the regularizing norm or seminorm, and γ ∊ (0, 1) is a robustness parameter. We show that R 1 GCV is less likely to choose a very small value of λ than both GCV and RGCV. RGCV and R 1 GCV also have good asymptotic properties for general problems with independent errors. Strengthening previous results for RGCV, it is shown that the (shifted) RGCV and R 1 GCV functions are consistent estimates of 'robust risk' functions, which place extra weight on the variance of f λ . In addition RGCV is asymptotically equivalent to the modified GCV method. The results of numerical simulations for R 1 GCV are consistent with the asymptotic results, and, for suitable values of γ, R 1 GCV is more reliable and accurate than GCV.