Bootstrap, wild bootstrap, and asymptotic normality

Abstract

We show for an i.i.d. sample that bootstrap estimates consistently the distribution of a linear statistic if and only if the normal approximation with estimated variance works. An asymptotic approach is used where everything may depend on n. The result is extended to the case of independent, but not necessarily identically distributed random variables. Furthermore it is shown that wild bootstrap works under the same conditions as bootstrap. © 1992 Springer-Verlag.