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A generalized approximate cross validation for smoothing splines with non-Gaussian data. (English) Zbl 0854.62044
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.

62G07Density estimation
65C99Probabilistic methods, simulation and stochastic differential equations (numerical analysis)