  Prediction error is critical to assessing the performance of statistical
methods and selecting statistical models. We propose the cross-validation and
approximated cross-validation methods for estimating prediction error under a
broad q-class of Bregman divergence for error measures which embeds nearly all
of the commonly used loss functions in regression, classification procedures
and machine learning literature. The approximated cross-validation formulas are
analytically derived, which facilitate fast estimation of prediction error
under the Bregman divergence. We then study a data-driven optimal bandwidth
selector for the local-likelihood estimation that minimizes the overall
prediction error or equivalently the covariance penalty. It is shown that the
covariance penalty and cross-validation methods converge to the same
mean-prediction-error-criterion. We also propose a lower-bound scheme for
computing the local logistic regression estimates and demonstrate that it is as
simple and stable as the local least-squares regression estimation. The
algorithm monotonically enhances the target local-likelihood and converges. The
idea and methods are extended to the generalized varying-coefficient models and
semiparametric models.
