Summary: 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-out-one 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 \(\lambda\) chosen by this approximation procedure with the \(\lambda\) chosen from the two most often used algorithms based on the generalized cross validation procedure. In the examples here, the GACV estimate produces a better fit of the truth in terms of minizing 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.