Hybrid bootstrap procedures for shrinkage-type estimators

Abstract

In statistical inference, one is often interested in estimating the distribution of a root, which is a function of the data and the parameters only. Knowledge of the distribution of a root is useful for inference problems such as hypothesis testing and the construction of a confidence set. Shrinkage-type estimators have become popular in statistical inference due to their smaller mean squared errors. In this thesis, the performance of different bootstrap methods is investigated for estimating the distributions of roots which are constructed based on shrinkage estimators. Focus is on two shrinkage estimation problems, namely the James-Stein estimation and the model selection problem in simple linear regression. A hybrid bootstrap procedure and a bootstrap test method are proposed to estimate the distributions of the roots of interest. In the two shrinkage problems, the asymptotic errors of the traditional n-out-of-n bootstrap, m-out-of-n bootstrap and the proposed methods are derived under a moving parameter framework. The problem of the lack of uniform consistency of the n-out-of-n and the m-out-of-n bootstraps is exposed. It is shown that the proposed methods have better overall performance, in the sense that they yield improved convergence rates over almost the whole range of possible values of the underlying parameters. Simulation studies are carried out to illustrate the theoretical findings.