By Kenneth B Stolarsky

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Thus, the bootstrapped estimators have different (conditional) means under the two methods. However, note that if the process {Xn h> 1 satisfies some standard moment and mixing conditions, then E{E*(O:n~) - E*O~:Jp = O(Cjn2). Hence the difference between the two is negligible for large sample sizes. 7 Generalized Block Bootstrap As follows from its description (cf. 5), the MBB resampling scheme suffers from an undesirable boundary effect as it assigns lesser 32 2. Bootstrap Methods weights to the observations toward the beginning and the end of the data set than to the middle part.

1 Consistency of Bootstrap Variance Estimators The bootstrap variance estimators Var*(T~(j)), j = 1,2,3 have a very desirable property, namely, that they can be expressed by simple, closedform formulas involving the observations X n , and thus, may be computed directly without any Monte-Carlo simulations. This is possible because of the linearity of the bootstrap sample mean in the resampled observations. Let Ui = (Xi + ... + XiH-I)/£ denote the average of the block (Xi, ... ) == U(i-1)£+1 = (X(i-l)£+l + ...

Next draw a simple random sample E;+l' ... 8), as: xt = Xi for i = 1, ... ,p, and 24 2. Bootstrap Methods <:, Note that by construction p < i ~ mare iid and E*Ei = O. The bootstrap version of the random variable Tn = tn(Xn; F, (3) is defined as where X,';, = {X;, ... , X,';,} and Fn denotes the empirical distribution of the centered residuals Ei, p < i ~ n. The sampling distribution of Tn is approximated by the conditional distribution of T,';, n given X n . 8), different versio~s of this resampling scheme have been proposed by Freedman (1984), Efron and Tibshirani (1986), Swanepoel and van Wyk (1986), and Kreiss and Franke (1992).

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Algebraic numbers and Diophantine approximation by Kenneth B Stolarsky
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