In a regression problem, typically there are p explanatory variables possibly related to a response variable, and we wish to select a subset of the p explanatory variables to fit a model between these variables and the response. A bootstrap variable/model selection procedure is to select the subset of variables by minimizing bootstrap estimates of the prediction error, where the bootstrap estimates are constructed based on a data set of size n. Although the bootstrap estimates have good properties, this bootstrap selection procedure is inconsistent in the sense that the probability of selecting the optimal subset of variables does not converge to 1 as n → ∞. This inconsistency can be rectified by modifying the sampling method used in drawing bootstrap observations. For bootstrapping pairs (response, explanatory variable), it is found that instead of drawing n bootstrap observations (a customary bootstrap sampling plan), much less bootstrap observations should be sampled: The bootstrap selection p...

Bootstrap Model Selection

ummary
A dynamic treatment regime consists of a set of decision rules that dictate how to individualize treatment to patients based on available treatment and covariate history. A common method for estimating an optimal dynamic treatment regime from data is Q-learning which involves nonsmooth operations of the data. This nonsmoothness causes standard asymptotic approaches for inference like the bootstrap or Taylor series arguments to breakdown if applied without correction. Here, we consider the m-out-of-n bootstrap for constructing confidence intervals for the parameters indexing the optimal dynamic regime. We propose an adaptive choice of m and show that it produces asymptotically correct confidence sets under fixed alternatives. Furthermore, the proposed method has the advantage of being conceptually and computationally much simple than competing methods possessing this same theoretical property. We provide an extensive simulation study to compare the proposed method with currently available inference procedures. The results suggest that the proposed method delivers nominal coverage while being less conservative than alternatives. The proposed methods are implemented in the qLearn R-package and have been made available on the Comprehensive R-Archive Network (http://cran.r-project.org/). Analysis of the Sequenced Treatment Alternatives to Relieve Depression (STAR*D) study is used as an illustrative example.

Inference for Optimal Dynamic Treatment Regimes Using an Adaptive m-Out-of-n Bootstrap Scheme

An Edgeworth expansion for the m out of n bootstrapped median

We consider the asymptotic behaviour of L1 -estimators in a linear regression under a very general form of heteroscedasticity. The limiting distributions of the estimators are derived under standard conditions on the design. We also consider the asymptotic behaviour of the bootstrap in the heteroscedastic model and show that it is consistent to first order only if the limiting distribution is normal.



L'auteur etudie le comportement asymptotique d'estimateurs L1 dans le cadre de la regression lineaire, sous des hypotheses d'heteroscedasticite tres generates. Il determine leur loi limite dans des conditions classiques portant sur la matrice d'incidence. Il examine en outre le comportement asymptotique du bootstrap dans le modele heteroscedastique et montre que la convergence du premier ordre ne se produit que si la loi limite est gaussienne.

J. S. Huang

Papers

Bootstrapping a sample quantile when the density has a jump