Estimation in the Mixture Transition Distribution Model for High Order Markov Chains

Abstract : The mixture transition distribution (MTD) model was introduced by Raftery (1985) as a parsimonious model for high-order Markov chains. It is flexible, can represent a wide range of dependence patterns, can be physically motivated, fits data well, and appears to be a discrete-valued analogue for the class of autoregressive time series models. However, estimation has presented difficulties because the parameter space is highly nonconvex, being defined by a large number of nonlinear constraints. Here we propose an efficient computational algorithm for maximum likelihood estimation which is based on a way of reducing the large number of constraints. This also allows more structured versions of the model, for example those involving structural zeros, to be fit quite easily. A way of fitting the model using GLIM is also discussed. The algorithm is applied to a sequence of wind directions, and also to two sequences of DNA bases from introns from genes of the mouse. In each case, the MTD model fits better than the conventional Markov chain model.

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