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Bootstrap choice of tuning parameters
Authors:Christian Léger  Joseph P Romano
Institution:(1) Département d'informatique et Recherche Opérationnelle, Université de Montréal, Succursale A, CP 6128, H3C 3J7 Montréal, Québec, Canada;(2) Department of Statistics, Stanford University, 94305 Stanford, CA, USA
Abstract:Consider the problem of estimating θ=θ(P) based on datax n from an unknown distributionP. Given a family of estimatorsT n, β of θ(P), the goal is to choose β among β∈I so that the resulting estimator is as good as possible. Typically, β can be regarded as a tuning or smoothing parameter, and proper choice of β is essential for good performance ofT n, β . In this paper, we discuss the theory of β being chosen by the bootstrap. Specifically, the bootstrap estimate of β, 
$$\hat \beta _n$$
, is chosen to minimize an empirical bootstrap estimate of risk. A general theory is presented to establish the consistency and weak convergence properties of these estimators. Confidence intervals for θ(P) based on 
$$T_{n,\hat \beta _n }$$
, are also asymptotically valid. Several applications of the theory are presented, including optimal choice of trimming proportion, bandwidth selection in density estimation and optimal combinations of estimates.
Keywords:Bandwidth selection  bootstrap  confidence limits  density estimation  risk function
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