Inference for Stochastically Contaminated Variable Length Markov Chains

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.