A theoretical analysis for both fixed design and random design settings is provided. The estimators admit an explicit form and can be linked to LASSO and Principal Component Regression (PCR). We propose a new family of estimators, called the canonical thresholding estimators, which pick largest regression coefficients in the canonical form. Unlike many papers on the topic, we do not require sparsity of the regression coefficients instead, our main structural assumption is a decay of eigenvalues of the covariance matrix of the data. Numerical simulations confirm good performance of the proposed estimators compared to the previously developed methods.Ībstract = "We consider a high-dimensional linear regression problem. Some minimax lower bounds are established to showcase the optimality of our procedure. The study of these relative errors leads to a new concept of joint effective dimension, which incorporates the covariance of the data and the regression coefficients simultaneously, and describes the complexity of a linear regression problem. In addition, we promote the use of the relative errors, strongly linked with the out-of-sample R2. Obtained bounds on the mean squared error and the prediction error of a specific estimator from the family allow to clearly state sufficient conditions on the decay of eigenvalues to ensure convergence. We consider a high-dimensional linear regression problem.
0 Comments
Leave a Reply. |
AuthorWrite something about yourself. No need to be fancy, just an overview. ArchivesCategories |