On robustness of model-based bootstrap schemes in nonparametric time series analysis

Theory in time series analysis is often developed in the context of finite-dimensional models for the data generating process. Whereas corresponding estimators such as those of a conditional mean function are reasonable even if the true dependence mechanism is of a more complex structure, it is usually necessary to capture the whole dependence structure asymptotically for the bootstrap to be valid. However, certain model-based bootstrap methods remain valid for some interesting quantities arising in nonparametric statistics. We generalize the well-known whitening by windowing principle to joint distributions of nonparametric estimators of the autoregression function. As a consequence, we obtain that model-based nonparametric bootstrap schemes remain valid for supremum-type functionals as long as they mimic the corresponding finite-dimensional joint distributions consistently. As an example, we investigate a finite order Markov chain bootstrap in the context of a general stationary process.