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Realistic Predictive Risk: The Role of Penalty and Covariate Diffusion in Model Selection

Jiang, Jieyi, Jiang

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2017, Doctor of Philosophy, Ohio State University, Statistics.
One important goal of model selection is the minimization of predictive risk. First, fold-wise cross-validation aims at estimating the predictive risk and identifying the structure of the linear model. However, it leads to inconsistent model selection and is inclined to select overfitting models asymptotically. Current regularized regression methods use a penalty in conjunction with a loss function for model fitting. A penalty can be used not only for model fitting but also for model evaluation. We propose the Penalized Cross-Validation Criterion: a suitable penalty term is added to the cross-validation score to ensure consistent model selection. With squared error loss, we find sufficient conditions on the penalty for consistency, and propose a suitable penalty for samples of finite size. We extend the result to negative Gaussian log-likelihood. Simulation studies show the advantage of penalized cross-validation in model selection. The second part of this dissertation discusses the change in predictive risk due to diffused covariates. Typical derivations for model selection criteria assume that the distributions of covariates in the training set and future-prediction set are identical. In practice, we are often most interested in forecasts over a novel covariate distribution. This difference in intended use of the model impacts predictive risk, estimates of predictive risk, and summaries arising from it. We formulate the predictive problem with divergence of the future-prediction set from the training set in terms of a diffusion of the distribution of covariates. As the covariates in the training set are diffused, measures of model complexity based on predictive risk change, impacting both model selection and choice of tuning parameters. We provide results in several settings, including subset selection, polynomial regression and ridge regression, and propose ways to adjust the selection of model and tuning parameter. Simulation studies show the benefits of the adjustment.
Steven MacEachern (Advisor)
Yoonkyung Lee (Advisor)
Xinyi Xu (Committee Member)
Yunzhang Zhu (Committee Member)

Recommended Citations

Citations

  • Jiang, Jiang, J. (2017). Realistic Predictive Risk: The Role of Penalty and Covariate Diffusion in Model Selection [Doctoral dissertation, Ohio State University]. OhioLINK Electronic Theses and Dissertations Center. http://rave.ohiolink.edu/etdc/view?acc_num=osu1503072235693181

    APA Style (7th edition)

  • Jiang, Jiang, Jieyi. Realistic Predictive Risk: The Role of Penalty and Covariate Diffusion in Model Selection. 2017. Ohio State University, Doctoral dissertation. OhioLINK Electronic Theses and Dissertations Center, http://rave.ohiolink.edu/etdc/view?acc_num=osu1503072235693181.

    MLA Style (8th edition)

  • Jiang, Jiang, Jieyi. "Realistic Predictive Risk: The Role of Penalty and Covariate Diffusion in Model Selection." Doctoral dissertation, Ohio State University, 2017. http://rave.ohiolink.edu/etdc/view?acc_num=osu1503072235693181

    Chicago Manual of Style (17th edition)