The asymptotic distribution of the isotonic regression estimator over a countable pre-ordered set

We study the isotonic regression estimator over a general countable pre-ordered set. We obtain the limiting distribution of the estimator and study its properties: It is proved that, under some general assumptions, the limiting distribution of the isotonised estimator is given by the concatenation of the separate isotonic regressions of the restrictions of an underlying estimator's asymptotic distribution to the comparable level sets of the underlying estimator's probability limit. Also, we show that the isotonisation preserves the rate of convergence of the underlying estimator. We apply these results to the problems of estimation of a bimonotone regression function and estimation of a bimonotone probability mass function.