ABSTRACT
Note that we have used the centering H(X11 ) in the definitions of the bootstrapped statistics T,!* and Ti,, which is apparently different from the definitions of the bootstrap versions of r,t and T11 under other block bootstrap methods. It is known (cf. Lahiri (1992)) that for the existing block bootstrap methods based on nonrandom block lengths, centering H(Xi,) at H(X11 ) in the definition of a bootstrapped statistic does not generally yield second-order accurate approximations. Because of the nonstationarity of the bootstrap observations under these resampling methods, the conditional expectation of the bootstrap sample mean is usually different from X11 , and therefore, the natural choice of centering H(X11 ) does not work. Consequently, one needs to modify the definitions of bootstrapped statistics suitably to attain higher order accuracy. An advantage of the SB method is that no such additional tuning of the bootstrapped statistics T,!* and Ti, is required. The conditional expectation of Xi, under the SB is X11 for all 11, and the natural choice of the centering can be used for applying the SB. Indeed, in the next Section we show that with the above definitions, the conditional distributions of T,!* and Ti, (given the data) provide secondorder accurate approximations to the unknown sampling distributions of the asymptotically pivotal quantities T,; and T11 •
ASSUMPTIONS
(A.2) There exists {j > 0 such that for all n, k = I, 2, ... with k > 5-1, there exists a ~;;~t-measurable random vector X11 .k such that
