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

Theory in time series analysis is often developed in the context of nite-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 estima-tors 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 nite-dimensional joint distributions consistently. As an example, we investigate a nite order Markov chain bootstrap in the context of a general stationary process. 1. Introduction One of the major merits of the bootstrap is its universality: it is valid for a variety of diierent purposes (statistics) and under quite general assumptions on the distributions of the observations. For i.i.d. data, it is easy to implement and usually one does not need severe conditions for its validity. Without the assumption of independence of the observations, the construction of valid resampling schemes becomes more diicult since one has to appropriately mimic the dependence mechanism. Also in this context, there exist nearly assumption-free methods. Hall (1985), Carlstein (1986) and Shi (1986) proposed resampling from nonoverlapping blocks of increasing length which was later reened by K unsch (1989). Other modiications are the circular block bootstrap proposed by Politis and On the other hand, there exists an extensive literature on model-based bootstrap methods in the time series context. Under the assumption of i.i.d. innovations in a linear autoregressive model, Efron and Tibshirani (1986) proposed to generate bootstrap series by drawing bootstrap innovations independently with replacement from the set of mean-adjusted residuals. Kreiss and Franke (1992) generalized this to autoregressive moving average models. Furthermore, there exists a series of proposals for bootstrapping Markov chains; see the brief survey in Section 3. There also exist several semiparametric methods. For example, Kreiss (1988) approximated linear autoregressive processes by a bootstrap process of nite, but increasing order. Franke and Wendel (1992) and Kreutzberger (1993) generalized the method of Efron and Tibshirani (1986) to the case of nonlinear autoregressive processes.