Statistical analysis of parsimonious high-order multivariate finite Markov chains based on sufficient statistics

Abstract

A new parsimonious $\mathrm{MCSS}\left(s\right)$ (which stands for “Markov Chain of order $s$ based on Sufficient Statistics”) model for multivariate discrete-valued time series is constructed. The $\mathrm{MCSS}\left(s\right)$ model has sufficient statistics of a simple form based on multivariate frequencies of $(s+1)$-tuples for observed time series. Special cases of the $\mathrm{MCSS}\left(s\right)$ model and their relations to the results known in the literature are discussed. The strong concavity property of the loglikelihood function and the uniqueness of the maximum likelihood estimator under mild regularity conditions are proven for the $\mathrm{MCSS}\left(s\right)$ model. Forecasting statistics for the multivariate discrete-valued time series derived with the $\mathrm{MCSS}\left(s\right)$ model are constructed. The developed theory is illustrated with computer experiments on simulated and real data.