Comparing six shrinkage estimators with large sample theory and asymptotically optimal prediction intervals

• Published in: Statistical papers (Berlin, Germany) Vol. 62; no. 5; pp. 2407 - 2431
• Main Authors: Watagoda, Lasanthi C. R. Pelawa, Olive, David J.
• Format: Journal Article
• Language: English
• Published: Berlin/Heidelberg Springer Berlin Heidelberg 01.10.2021; Springer Nature B.V
• Subjects: Asymptotic properties; Economic Theory/Quantitative Economics/Mathematical Methods; Economics; Estimators; Finance; Insurance; Intervals; Management; Mathematics and Statistics; Operations Research/Decision Theory; Probability Theory and Stochastic Processes; Random variables; Regression analysis; Regression models; Research Article; Shrinkage; Statistics; Statistics for Business; Ridge regression; Partial least squares; Lasso; Principal components regression; Forward Selection
• ISSN: 0932-5026; 1613-9798
• DOI: 10.1007/s00362-020-01193-1

Consider the multiple linear regression model Y = β 1 + β 2 x 2 + ⋯ + β p x p + e = x T β + e with sample size n . This paper compares the six shrinkage estimators: forward selection, lasso, partial least squares, principal components regression, lasso variable selection, and ridge regression, with large sample theory and two new prediction intervals that are asymptotically optimal if the estimator β ^ is a consistent estimator of β . Few prediction intervals have been developed for p > n , and they are not asymptotically optimal. For p fixed, the large sample theory for variable selection estimators like forward selection is new, and the theory shows that lasso variable selection is n consistent under much milder conditions than lasso. This paper also simplifies the proofs of the large sample theory for lasso, ridge regression, and elastic net.