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.
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Léger, C., Romano, J.P. Bootstrap choice of tuning parameters. Ann Inst Stat Math 42, 709–735 (1990). https://doi.org/10.1007/BF02481146
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DOI: https://doi.org/10.1007/BF02481146