A Generalized Approximate Cross Validation for Smoothing Splines with Non-gaussian Data

In this paper, we propose a Generalized Approximate Cross Validation (GACV) function for estimating the smoothing parameter in the penalized log likelihood regression problem with non-Gaussian data. This GACV is obtained by, first, obtaining an approximation to the leaving-out-one function based on the negative log likelihood, and then, in a step reminiscent of that used to get from leaving-outone cross validation to GCV in the Gaussian case, we replace diagonal elements of certain matrices by 1/n times the trace. A numerical simulation with Bernoulli data is used to compare the smoothing parameter λ chosen by this approximation procedure with the λ chosen from the two most often used algorithms based on the generalized cross validation procedure (O’Sullivan et al. (1986), Gu (1990, 1992)). In the examples here, the GACV estimate produces a better fit of the truth in term of minimizing the Kullback-Leibler distance. Figures suggest that the GACV curve may be an approximately unbiased estimate of the Kullback-Leibler distance in the Bernoulli data case; however, a theoretical proof is yet to be found.