The block bootstrap approximates sampling distributions from dependent data by resampling data blocks. A fundamental problem is establishing its consistency for the distribution of a sample mean, as a prototypical statistic. We use a structural relationship with subsampling to characterize the bootstrap in a new and general manner. While subsampling and block bootstrap differ, the block bootstrap distribution of a sample mean equals that of a k-fold self-convolution of a subsampling distribution. Motivated by this, we provide simple necessary and sufficient conditions for a convolved subsampling estimator to produce a normal limit that matches the target of bootstrap estimation. These conditions may be linked to consistency properties of an original subsampling distribution, which are often obtainable under minimal assumptions. Through several examples, the results are shown to validate the block bootstrap for means under significantly weakened assumptions in many existing (and some new) dependence settings, which also addresses a standing conjecture of Politis, Romano and Wolf (1999). Beyond sample means, the convolved subsampling estimator may not match the block bootstrap, but instead provides a hybrid-resampling estimator of interest in its own right. For general statistics with normal limits, results also establish the consistency of convolved subsampling under minimal dependence conditions, including non-stationarity.
Bootstrap for nonlinear statistics like U-statistics of dependent data has been studied by several authors. This is typically done by producing a bootstrap version of the sample and plugging it into the statistic. We suggest an alternative approach of getting a bootstrap version of U-statistics. We will show the consistency of the new method and compare its finite sample properties in a simulation study and by applying both methods to financial data.
We consider the change-point problem for the marginal distribution of subordinated Gaussian processes that exhibit long-range dependence. The asymptotic distributions of Kolmogorov-Smirnov-and Cramér-von Mises type statistics are investigated under local alternatives. By doing so we are able to compute the asymptotic relative efficiency of the mentioned tests and the CUSUM test. In the special case of a mean-shift in Gaussian data it is always 1. Moreover our theory covers the scenario where the Hermite rank of the underlying process changes.In a small simulation study we show that the theoretical findings carry over to the finite sample performance of the tests..Keywords: asymptotic relative efficiency, change-point test, empirical process, local alternatives, long-range dependence.where the convergence takes place in D([0, 1] × [−∞, ∞]), equipped with the uniform topology.and (Z m,H (t)) t∈[0,1] is an m-th order Hermite process, see Taqqu (1979) for a definition.Remark 2.1. In the case m = 1, the Hermite process becomes the well known fractional Brownian Motion, which we denote by B H (t).
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