Consider $d$ dependent change point tests, each based on a CUSUM-statistic. We provide an asymptotic theory that allows us to deal with the maximum over all test statistics as both the sample size $n$ and $d$ tend to infinity. We achieve this either by a consistent bootstrap or an appropriate limit distribution. This allows for the construction of simultaneous confidence bands for dependent change point tests, and explicitly allows us to determine the location of the change both in time and coordinates in high-dimensional time series. If the underlying data has sample size greater or equal $n$ for each test, our conditions explicitly allow for the large $d$ small $n$ situation, that is, where $n/d\to0$. The setup for the high-dimensional time series is based on a general weak dependence concept. The conditions are very flexible and include many popular multivariate linear and nonlinear models from the literature, such as ARMA, GARCH and related models. The construction of the tests is completely nonparametric, difficulties associated with parametric model selection, model fitting and parameter estimation are avoided. Among other things, the limit distribution for $\max_{1\leq h\leq d}\sup_{0\leq t\leq1}\vert \mathcal{W}_{t,h}-t\mathcal{W}_{1,h}\vert$ is established, where $\{\mathcal{W}_{t,h}\}_{1\leq h\leq d}$ denotes a sequence of dependent Brownian motions. As an application, we analyze all S&P 500 companies over a period of one year.Comment: Clarified Assumption 4.3 and Theorems 4.4, 4.6 and 4.8. The results themselves are unchange

This work develops change-point methods for statistics of high-frequency data. The main interest is the volatility of an Itô semi-martingale, which is discretely observed over a fixed time horizon. We construct a minimax-optimal test to discriminate different smoothness classes of the underlying stochastic volatility process. In a high-frequency framework we prove weak convergence of the test statistic under the hypothesis to an extreme value distribution. As a key example, under extremely mild smoothness assumptions on the stochastic volatility we thereby derive a consistent test for volatility jumps. A simulation study demonstrates the practical value in finite-sample applications.

We consider the model of nonregular nonparametric regression where smoothness constraints are imposed on the regression function f and the regression errors are assumed to decay with some sharpness level at their endpoints. The aim of this paper is to construct an adaptive estimator for the regression function f . In contrast to the standard model where local averaging is fruitful, the nonregular conditions require a substantial different treatment based on local extreme values. We study this model under the realistic setting in which both the smoothness degree β > 0 and the sharpness degree a ∈ (0, ∞) are unknown in advance. We construct adaptation procedures applying a nested version of Lepski's method and the negative Hill estimator which show no loss in the convergence rates with respect to the general L q -risk and a logarithmic loss with respect to the pointwise risk. Optimality of these rates is proved for a ∈ (0, ∞). Some numerical simulations and an application to real data are provided.

Let {X k } k≥Z be a stationary sequence. Given p ∈ (2, 3] moments and a mild weak dependence condition, we show a Berry-Esseen theorem with optimal rate n p/2−1 . For p ≥ 4, we also show a convergence rate of n 1/2 in L q -norm, where q ≥ 1. Up to log n factors, we also obtain nonuniform rates for any p > 2. This leads to new optimal results for many linear and nonlinear processes from the time series literature, but also includes examples from dynamical system theory. The proofs are based on a hybrid method of characteristic functions, coupling and conditioning arguments and ideal metrics.

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