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Summary Model selection is crucial both to high-dimensional learning and to inference for contemporary big data applications in pinpointing the best set of covariates among a sequence of candidate interpretable models. Most existing work implicitly assumes that the models are correctly specified or have fixed dimensionality, yet both model misspecification and high dimensionality are prevalent in practice. In this paper, we exploit the framework of model selection principles under the misspecified generalized linear models presented in Lv & Liu (2014), and investigate the asymptotic expansion of the posterior model probability in the setting of high-dimensional misspecified models. With a natural choice of prior probabilities that encourages interpretability and incorporates the Kullback–Leibler divergence, we suggest using the high-dimensional generalized Bayesian information criterion with prior probability for large-scale model selection with misspecification. Our new information criterion characterizes the impacts of both model misspecification and high dimensionality on model selection. We further establish the consistency of covariance contrast matrix estimation and the model selection consistency of the new information criterion in ultrahigh dimensions under some mild regularity conditions. Our numerical studies demonstrate that the proposed method enjoys improved model selection consistency over its main competitors.
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Model selection properties of forward selection and sequential cross‐validation for high‐dimensional regression
Forward selection (FS) is a popular variable selection method for linear regression. But theoretical understanding of FS with a diverging number of covariates is still limited. We derive sufficient conditions for FS to attain model selection consistency. Our conditions are similar to those for orthogonal matching pursuit, but are obtained using a different argument. When the true model size is unknown, we derive sufficient conditions for model selection consistency of FS with a data‐driven stopping rule, based on a sequential variant of cross‐validation. As a byproduct of our proofs, we also have a sharp (sufficient and almost necessary) condition for model selection consistency of “wrapper” forward search for linear regression. We illustrate intuition and demonstrate performance of our methods using simulation studies and real datasets.
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- Award ID(s):
- 2015492
- PAR ID:
- 10367386
- Publisher / Repository:
- Wiley Blackwell (John Wiley & Sons)
- Date Published:
- Journal Name:
- Canadian Journal of Statistics
- Volume:
- 50
- Issue:
- 2
- ISSN:
- 0319-5724
- Page Range / eLocation ID:
- p. 454-470
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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