Grenander--Stone estimator: stacked constrained estimation of a discrete distribution over a general directed acyclic graph

In this paper we integrate isotonic regression with Stone's cross-validation-based method to estimate a distribution with a general countable support with a partial order relation defined on it. We prove that the estimator is strongly consistent for any underlying distribution, derive its rate of convergence, and in the case of one-dimensional support we obtain Marshal-type inequality for cumulative distribution function of the estimator. Also, we construct the asymptotically correct conservative global confidence band for the estimator. It is shown that, first, the estimator performs good even for small sized data sets, second, the estimator outperforms in the case of non-isotonic underlying distribution, and, third, it performs almost as good as Grenander estimator when the true distribution is isotonic. Therefore, the new estimator provides a trade-off between goodness-of-fit, monotonicity and quality of probabilistic forecast. We apply the estimator to the time-to-onset data of visceral leishmaniasis in Brazil collected from $2007$ to $2014$.