Higher — order accuracy of bootstrap for smooth functionals

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).