Anne Leucht

Dependent wild bootstrap for degenerate U- and V-statistics

Degenerate U- and V-statistics play an important role in the field of hypothesis testing since numerous test statistics can be formulated in terms of these quantities. Therefore, consistent bootstrap methods for U- and V-statistics can be applied in order to determine critical values for these tests. We prove a new asymptotic result for degenerate U- and V-statistics of weakly dependent random variables. As our main contribution, we propose a new model-free bootstrap method for U- and V-statistics of dependent random variables. Our method is a modification of the dependent wild bootstrap recently proposed by Shao [X. Shao, The dependent wild bootstrap, J. Amer. Statist. Assoc. 105 (2010) 218-235], where we do not directly bootstrap the underlying random variables but the summands of the U- and V-statistics. Asymptotic theory for the original and bootstrap statistics is derived under simple and easily verifiable conditions. We discuss applications to a Cramer-von Mises-type test and a two sample test for the marginal distribution of a time series in detail. The finite sample behavior of the Cramer-von Mises test is explored in a small simulation study. While the empirical size was reasonably close to the nominal one, we obtained nontrivial empirical power in all cases considered.

Degenerate U- and V-statistics under weak dependence: Asymptotic theory and bootstrap consistency

Numerous test statistics can be approximated by statistics of U- or V-type. In the case of i.i.d. random variables the limit distribution can be derived by a spectral decomposition of the kernel if the latter is square integrable. To use the same method for dependent data, restrictive assumptions on the associated eigenvalues and eigenfunctions are required. In the majority of cases, it is quite complicated or even impossible to check these conditions. Therefore we employ a wavelet decomposition of the kernel in order to derive the asymptotic distributions of U- and V-statistics for weakly dependent data. This approach only requires some moment constraints and smoothness assumptions concerning the kernel. The asymptotic distributions of U- and V-statistics for both independent and weakly dependent observations cannot be used directly since they depend on certain parameters, which in turn depend on the underlying situation in a complicated way. Therefore, problems arise as soon as critical values for test statistics of U- and V-type have to be determined. The bootstrap offers a convenient way to circumvent these problems, see [3] for the i.i.d. case. We derive the consistency of general bootstrap methods for statistics of weakly dependent data. The results are then applied to construct bootstrap-based hypothesis tests of L2-type for weakly dependent data such as a model specification test and tests for symmetry or the parametric class of the marginal distribution of a time series.

Consistency of general bootstrap methods for degenerate U-type and V-type statistics

We provide general results on the consistency of certain bootstrap methods applied to degree-2 degenerate statistics of U-type and V-type. While it follows from well known results that the original statistic converges in distribution to a weighted sum of centred chi-squared random variables, we use a coupling idea of Dehling and Mikosch to show that the bootstrap counterpart converges to the same distribution. The result is applied to a goodness-of-fit test based on the empirical characteristic function.

Dependent Wild Bootstrap for the Empirical Process

In this paper, we propose a model-free bootstrap method for the empirical process under absolute regularity. More precisely, consistency of an adapted version of the so-called dependent wild bootstrap, that was introduced by Shao (2010) and is very easy to implement, is proved under minimal conditions on the tuning parameter of the procedure. We apply our results to construct confidence intervals for unknown parameters and to approximate critical values for statistical tests. A simulation study shows that our method is competitive to standard block bootstrap methods in finite samples.

Tests Based on Simplicial Depth for AR(1) Models With Explosion

We propose outlier a robust and distribution‐free test for the explosive AR(1) model with intercept based on simplicial depth. In this model, simplicial depth reduces to counting the cases where three residuals have alternating signs. The asymptotic distribution of the test statistic is given by a specific Gaussian process. Conditions for the consistency are given, and the power of the test at finite samples is compared with alternative tests. The new test outperforms these tests in the case of skewed errors and outliers. Finally, we apply the method to crack growth data and compare the results with an OLS approach.