ABSTRACT
A limitation of the preceding two approaches is that they apply only to linear statistics and cannot be easily extended to more general statistics such as the sample quantiles. A third and more general approach is to treat pivots 'if~ and rr'N as random signed measures and study their convergence in the functional sense. In this functional setting, we have established the following key results ([ll]). Theorem 2.1 Let F be a distribution function in R'. Let F~ and FN denote empirical distribution functions based respectively on the usual and the sequential bootstrap samples. Then
(3.1)
(3.3)
(3.4) N = 0:1 + O:z + · .. + O:n
in which a:1 , ... , O:n are iid Poisson random variable with mean .\ = 1 and with the added restriction that exactly m of the o:'s are non-zero, i.e. I{a 1 >0} + ... + I{an>O} = m. Further it can be shown that the moment generating function MN(t) of N is:
Let {~ : j 2 1} be a sequence of iid Poisson random variables with mean ). = 1. Let
(3.7)
(3.8)
(3.9)
(3.10)
(3.11)
(3.12)
(3.13)
(3.14)
- N Y=- n 1 n
For a real-valued function h, let
(3.16)
(3.17)
where ¢ denotes the standard normal density. In terms of the foregoing terminology, the following results furnish
a rigorous justification for the second order correctness of the sequential bootstrap. These results are consequences of a technical result on conditional Edgeworth expansions for weighted means of multivariate random vectors presented in [2]. Theorem 3.1 Let Xt, x2 , •.• be i.i.d. random vectors with mean p, and covariance matrix 2:. Let H be a 3-times continuously differentiable functwn in a neighborhood of p,. Suppose that x 1 has a strongly non-lattice
term correction captures the skewness of the underlying distribution. For further details the reader is referred to [2].
