Title: Inference for Stochastically Contaminated Variable Length Markov Chains,
Abstract: In this paper, we present a methodology to estimate the parameters of
stochastically contaminated models under two contamination regimes. In both
regimes, we assume that the original process is a variable length Markov chain
that is contaminated by a random noise. In the first regime we consider that
the random noise is added to the original source and in the second regime, the
random noise is multiplied by the original source. Given a contaminated sample
of these models, the original process is hidden. Then we propose a two steps
estimator for the parameters of these models, that is, the probability
transitions and the noise parameter, and prove its consistency. The first step
is an adaptation of the Baum-Welch algorithm for Hidden Markov Models. This
step provides an estimate of a complete order $k$ Markov chain, where $k$ is
bigger than the order of the variable length Markov chain if it has finite
order and is a constant depending on the sample size if the hidden process has
infinite order. In the second estimation step, we propose a bootstrap Bayesian
Information Criterion, given a sample of the Markov chain estimated in the
first step, to obtain the variable length time dependence structure associated
with the hidden process. We present a simulation study showing that our
methodology is able to accurately recover the parameters of the models for a
reasonable interval of random noises.