Approximations for weighted bootstrap processes with an application

Let [beta]n(t) denote the weighted (smooth) bootstrap process of an empirical process. We show that the order of the best Gaussian approximation for [beta]n(t) is n-1/2 log n and we construct a sequence of approximating Brownian bridges achieving this rate. We also obtain an approximation for [beta]n(t) using a suitably chosen Kiefer process. The result is applied to detect a possible change in the distribution of independent observations.

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