Assessing prediction error of nonparametric regression and classification under Bregman divergence

Jianqing Fan Chunming Zhang

Statistics Theory and Methods mathscidoc:1912.43420

arXiv preprint math/0506028, 2005.6
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.
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@inproceedings{jianqing2005assessing,
  title={Assessing prediction error of nonparametric regression and classification under Bregman divergence},
  author={Jianqing Fan, and Chunming Zhang},
  url={http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20191221114323120819980},
  booktitle={arXiv preprint math/0506028},
  year={2005},
}
Jianqing Fan, and Chunming Zhang. Assessing prediction error of nonparametric regression and classification under Bregman divergence. 2005. In arXiv preprint math/0506028. http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20191221114323120819980.
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