ABSTRACT
Theorem 3.1 shows that the expansion for the bootstrapped stattstlc remains valid as long as the expected block length p-1 grows to infinity at least as fast as a positive power of the sample size n. This can be relaxed to a lower bound of the form "p-1 > (logn) 1+' for some c: > 0" under stronger moment conditions on X1. In the same vein, the upper bound on "p-h' can be relaxed to "p-1 = O(n(l-cJ/2) for some E E (0, 1)", provided certain higher order absolute moments of X 1 are finite (depending on the value of c:). Politis and Romano (1994) show that the optimal expected block length (minimizing the mean squared error (MSE)) for estimating the variance of the estimtor B11 satisfies p-1 = C0n 113 (1 + o(l)) as n---> oo where C0 > 0 is a constant. Thus, the expansion for the bootstrapped statistic T~ remains valid for such optimal values of the blocking parameter p solely under the present set of conditions. The MSE optimal values of block lengths for estimating distribution functions are typically of a smaller order (cf. Hall, Horowitz and Jing ( 1995)) and hence, the conditions presented in the chapter cover such values of p as well.
