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In this paper we integrate isotonic regression with Stone'scross-validation-based method to estimate a distribution with a generalcountable support with a partial order relation defined on it. We prove thatthe estimator is strongly consistent for any underlying distribution, deriveits rate of convergence, and in the case of one-dimensional support we obtainMarshal-type inequality for cumulative distribution function of the estimator.Also, we construct the asymptotically correct conservative global confidenceband for the estimator. It is shown that, first, the estimator performs goodeven for small sized data sets, second, the estimator outperforms in the caseof non-isotonic underlying distribution, and, third, it performs almost as goodas Grenander estimator when the true distribution is isotonic. Therefore, thenew estimator provides a trade-off between goodness-of-fit, monotonicity andquality of probabilistic forecast. We apply the estimator to the time-to-onsetdata of visceral leishmaniasis in Brazil collected from $2007$ to $2014$.

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