Semibinomial conditionally nonlinear autoregressive models of discrete random sequences: probabilistic properties and statistical parameter estimation

We introduce a new model $\mathscr{P}\text{-}\mathrm{CNAR}(s)$ of sequences of discrete random variables with long memory determined by semibinomial conditionally nonlinear autoregression of order $s\in\N$ with small number of parameters. Probabilistic properties of this model are studied. For parameters of the model $\mathscr{P}\text{-}\mathrm{CNAR}(s)$ a family of consistent asymptotically normal statistical FB-estimates is suggested and the existence of an efficient FB-estimate is proved. Computational advantages of FB-estimate w.r.t. maximum likelihood estimate are shown: less restrictive sufficient conditions for uniqueness, explicit form of FB-estimate, fast recursive computation algorithm under extension of the model $\mathscr{P}\text{-}\mathrm{CNAR}(s)$. Subfamily of “sparse” FB-estimates that use some subset of frequencies of $s$-tuples is constructed, the asymptotic variance minimization problem within this subfamily is solved.