A Markov chain of order $s$ with $r$ partial connections and statistical inference on its parameters

We consider a homogeneous Markov chain of order $s$ with $r$ partial connections for which the transition probabilities from any state to the next depend not on all $s$ preceding states but on $r$ selected states only. We construct statistical estimators of parameters and statistical tests for values of these parameters of the model and investigate their asymptotic properties. Algorithms to calculate these estimators are suggested. The results of computer simulations are also given.