Penalized variable selection with U-estimates.

U-estimates are defined as maximizers of objective functions that are U-statistics. As an alternative to M-estimates, U-estimates have been extensively used in linear regression, classification, survival analysis, and many other areas. They may rely on weaker data and model assumptions and be preferred over alternatives. In this article, we investigate penalized variable selection with U-estimates. We propose smooth approximations of the objective functions, which can greatly reduce computational cost without affecting asymptotic properties. We study penalized variable selection using penalties that have been well investigated with M-estimates, including the LASSO, adaptive LASSO, and bridge, and establish their asymptotic properties. Generically applicable computational algorithms are described. Performance of the penalized U-estimates is assessed using numerical studies.