Variable Length

We study estimation in the class of stationary variable length Markov chains (VLMC) on a nite space. The processes in this class are still Markovian of higher order, but with memory of variable length yielding a much bigger and structurally richer class of models than ordinary higher order Markov chains. From a more algo-rithmic view, the VLMC model class has attracted interest in information theory and machine learning but statistical properties have not been explored very much. Provided that good estimation is available, an additional structural richness of the model class enhances predictive power by nding a better trade-oo between model bias and variance and allows better structural description which can be of speciic interest. The latter is exempliied with some DNA data. A version of the tree-structured context algorithm, proposed by Rissanen (1983) in an information theoretical setup , is shown to have new good asymptotic properties for estimation in the class of VLMC's, even when the underlying model increases in dimensionality: consistent estimation of minimal state spaces and mixing properties of tted models are given. We also propose a new bootstrap scheme based on tted VLMC's. We show its validity for quite general stationary categorical time series and for a broad range of statistical procedures.