Bootstrap for Shrinkage-Type Estimators

Abstract:     
The Hodges-LeCam and Stein estimators are the best-known examples of shrinkage-type estimators that attempt to improve upon the maximum likelihood and ordinary least squares with respect to the quadratic loss. It is known that when the mean is zero, Efron’s nonparametric bootstrap of the Hodges-LeCam estimator is not consistent. Likewise, the nonparametric bootstrap of the Stein estimator is not consistent when all the means are equal. For both estimators, the bootstrap distribution is random asymptotically. In this paper I first show that bootstrap consistency can be restored if the null hypothesis about the mean(s) is imposed in the bootstrap data generating process, rather than in the test statistic itself. Second, I show that the new remedy, combined with a finite-sample bias correction, improves the finite-sample properties of the bootstrap distribution of the Hodges-LeCam statistic, in regions of the parameter space where convergence is nonuniform. Finally, I show that for post-model selection estimators that share the so-called oracle property, the nonparametric bootstrap conditional on the unrestricted model is preferred to the bootstrap conditional on the selected model.

Number of Pages in PDF File: 26

Keywords: bootstrap inconsistency, resampling, hypothesis testing, super-efficient, shrinkage estimator; Stein, Hodges-LeCam, LASSO, SCAD, Bridge, Ridge estimator, oracle, sparsity property, consistent post-model selection estimator, unrestricted, over-identified, under-identified model

JEL Classification: C150, C120, C130, C520, C140, C210, C220

working papers series