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Regenerative block-bootstrap for Markov chains. (English) Zbl 1125.62037

Summary: A specific bootstrap method is introduced for positive recurrent Markov chains, based on the regenerative method and the Nummelin splitting technique. This construction involves generating a sequence of approximate pseudo-renewal times for a Harris chain \(X\) from data \(X_1,\dots,X_n\) and the parameters of a minorization condition satisfied by its transition probability kernel and then applying a variant of the methodology proposed by S. Datta and W. P. McCormick [Can. J. Stat. 21, No. 2, 181–193 (1993; Zbl 0780.62063)] for bootstrapping additive functionals of the type \(n^{-1} \sum_{n=1}^n f(X_i)\) when the chain possesses an atom. This novel methodology mainly consists in dividing the sample path of the chain into data blocks corresponding to the successive visits to the atom and resampling the blocks until the (random) length of the reconstructed trajectory is at least \(n\), so as to mimic the renewal structure of the chain.
In the atomic case we prove that our method inherits the accuracy of the bootstrap in the independent and identically distributed case up to \(O_{\mathbb P}(n^{-1})\) under weak conditions. In the general (not necessarily stationary) case asymptotic validity for this resampling procedure is established, provided that a consistent estimator of the transition kernel may be computed. The second-order validity is obtained in the stationary case (up to a rate close to \(O_{\mathbb P}(n^{-1})\) for regular stationary chains). A data-driven method for choosing the parameters of the minorization condition is proposed and applications to specific Markovian models are discussed.

MSC:

62G09 Nonparametric statistical resampling methods
60J10 Markov chains (discrete-time Markov processes on discrete state spaces)
62E20 Asymptotic distribution theory in statistics
65C60 Computational problems in statistics (MSC2010)

Citations:

Zbl 0780.62063
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