Phylogenetic distances for neighbour dependent substitution processes

We consider models of nucleotidic substitution processes where the rate of substitution at a given site depends on the state of its neighbours. For a wide class of such nonreversible models, we show how to compute consistent, mathematically exact, estimators of the time elapsed between aligned sequences, for an ancestral sequence and a present one, and also for two present sequences. In both cases, we provide asymptotic confidence intervals, valid for nucleotidic sequences of finite length. We compute explicit formulas for the estimators and for their confidence intervals in the simplest nontrivial case, the Jukes-Cantor model with CpG influence.