Degenerate U- and V-statistics under weak dependence : Asymptotic theory and bootstrap consistency
By Anne Leucht. 2012
53 citations recorded by the source corpus.
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.
Subfields: Management Science and Operations Research, Statistics and Probability
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In Management Science and Operations Research
- Arthur Gretton
- Kacper P. Chwialkowski
- Michael H. Neumann
- Heiko Strathmann
- E. Rio
- Balaji Lakshminarayanan
- Denis Belomestny
In Statistics and Probability
- Kacper P. Chwialkowski
- Arthur Gretton
- Heiko Strathmann
- Anne Leucht
- Michael H. Neumann
- Dino Sejdinovic
- Bernhard Schölkopf
- Bharath K. Sriperumbudur
Citations in the corpus
- A Kernel Test of Goodness of Fit (2016, methodology): “We use Theorem 2.1 (Leucht, 2012) to see that, under the null hypothesis, f(Z1,n, · · · , Zt,n) converges to zero in probability.”
- Degenerate $$U$$- and $$V$$-statistics under ergodicity: asymptotics, bootstrap and applications in statistics (2013, methodology): “However, since he invoked empirical process theory in order to establish the asymptotic null distribution, he had to impose a condition on the conditional distributions which is not fulfilled e.g. by Poisson count processes considered in Sect.”