Bootstrap choice of tuning parameters |
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Authors: | Christian Léger Joseph P Romano |
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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 |
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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,
, 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
, 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. |
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Keywords: | Bandwidth selection bootstrap confidence limits density estimation risk function |
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