  In this paper we integrate the isotonic regression with Stone's
cross-validation-based method to estimate discrete infinitely supported
distribution. We prove that the estimator is strongly consistent, derive its
rate of convergence for any underlying distribution, and for one-dimensional
case we derive 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-monotone 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.
