Assessing prediction error of nonparametric regression and classification under Bregman divergence
Abstract
Prediction error is critical to assessing the performance of statistical methods and selecting statistical models. We propose the crossvalidation and approximated crossvalidation methods for estimating prediction error under a broad qclass 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 crossvalidation formulas are analytically derived, which facilitate fast estimation of prediction error under the Bregman divergence. We then study a datadriven optimal bandwidth selector for the locallikelihood estimation that minimizes the overall prediction error or equivalently the covariance penalty. It is shown that the covariance penalty and crossvalidation methods converge to the same meanpredictionerrorcriterion. We also propose a lowerbound scheme for computing the local logistic regression estimates and demonstrate that it is as simple and stable as the local leastsquares regression estimation. The algorithm monotonically enhances the target locallikelihood and converges. The idea and methods are extended to the generalized varyingcoefficient models and semiparametric models.
 Publication:

arXiv Mathematics eprints
 Pub Date:
 June 2005
 DOI:
 10.48550/arXiv.math/0506028
 arXiv:
 arXiv:math/0506028
 Bibcode:
 2005math......6028F
 Keywords:

 Mathematics  Statistics;
 62G05 62H30
 EPrint:
 38 pages, 8 figures