Pseudo-regenerative block-bootstrap for hidden Markov chains

This paper is devoted to extend the regenerative block-bootstrap (RBB) proposed in [1] for regenerative Markov chains to Hidden Markov Models {(X _n, Y _n)} _n∈ℕ. In the HMM setup, regeneration times of the underlying chain X (i.e. consecutive times at which it visits a given state), which are regeneration times for the bivariate chain (X, Y) as well, are not observable. The principle underlying the RBB extension consists in resampling the output by generating first a sequence of approximate regeneration times for X from data Y ⁽ⁿ⁾ = (Y ₁, ⋯, Y _n), by splitting up next Y ⁽ⁿ⁾ into data blocks corresponding to the pseudo-renewal times obtained and, eventually, by resampling the blocks until the (random) length of the reconstructed series is a least n. Beyond the algorithmic description of the resampling procedure, which we call "hidden regenerative block-bootstrap" (HRBB), its performance is evaluated on a simple simulation example.