Strong robust generalized cross-validation for choosing the regularization parameter

Let f(lambda) be the Tikhonov regularized solution of a linear inverse or smoothing problem with discrete noisy data y(i), i = 1,..., n. To choose lambda 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 lambda to be the minimizer of gamma V(lambda) + (1 -gamma) F(1)(lambda), where V (lambda) is the GCV function, F(1)(lambda) is a certain approximate total measure of the influence of each data point on f(lambda) with respect to the regularizing norm or seminorm, and gamma is an element of ( 0, 1) is a robustness parameter. We show that R(1)GCV is less likely to choose a very small value of lambda 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(lambda). 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 gamma, R(1)GCV is more reliable and accurate than GCV.