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Higher — order accuracy of bootstrap for smooth functionals

  • Enno Mammen
Part of the Lecture Notes in Statistics book series (LNS, volume 77)

Abstract

We come now back to bootstrap of smooth statistical functionals. In Chapter 1 we have studied under which conditions bootstrap of smooth functionals works. In this chapter we give a simple proof for the higher order accuracy of the bootstrap estimate for smooth functionals T. Under the assumption that T admits a higher order expansion we show that the expectation of a smooth function H of the studentized functional is estimated to order Op(n-1). For symmetric smooth functions H, the rate of convergence is Op(n-3/2). These results are in agreement with Hall (1986a, 1988, 1992), where these rates of convergence have been shown for functionals which are implicit or explicit functions of vector means (for a discussion of these rates see also Beran, 1987). The idea of the proof in this chapter is to make a direct comparison of the bootstrap estimate and the true distribution using the idea of Lindeberg’s proof of the central limit theorem. The proof does not use Edgeworth expansions. The results of this chapter are also contained in Mammen(1990).

Keywords

Central Limit Theorem Bootstrap Procedure Bootstrap Estimate Bootstrap Confidence Interval High Order Accuracy 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Springer-Verlag New York, Inc. 1992

Authors and Affiliations

  • Enno Mammen
    • 1
  1. 1.Institut für Angewandte MathematikUniversität HeidelbergHeidelbergGermany

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