Mathematics – Statistics Theory
Scientific paper
2012-01-30
Annals of Statistics 2011, Vol. 39, No. 4, 2103-2130
Mathematics
Statistics Theory
Published in at http://dx.doi.org/10.1214/11-AOS900 the Annals of Statistics (http://www.imstat.org/aos/) by the Institute of
Scientific paper
10.1214/11-AOS900
We explore the limits of the autoregressive (AR) sieve bootstrap, and show that its applicability extends well beyond the realm of linear time series as has been previously thought. In particular, for appropriate statistics, the AR-sieve bootstrap is valid for stationary processes possessing a general Wold-type autoregressive representation with respect to a white noise; in essence, this includes all stationary, purely nondeterministic processes, whose spectral density is everywhere positive. Our main theorem provides a simple and effective tool in assessing whether the AR-sieve bootstrap is asymptotically valid in any given situation. In effect, the large-sample distribution of the statistic in question must only depend on the first and second order moments of the process; prominent examples include the sample mean and the spectral density. As a counterexample, we show how the AR-sieve bootstrap is not always valid for the sample autocovariance even when the underlying process is linear.
Kreiss Jens-Peter
Paparoditis Efstathios
Politis Dimitris N.
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