Isotonic estimation involves the estimation of a function which is known to be increasing with respect to a specified partial order. For the case of a linear order, a general theorem is given which simplifies and extends the techniques of Prakasa Rao and Brunk. Sufficient conditions for a specified limit distribution to obtain are expressed in terms of a local condition and a global condition. It is shown that the rate of convergence depends on the order of the first non-zero derivative and that this result can obtain even if the function is not monotone over its entire domain. The theorem is applied to give the asymptotic distributions of several estimators.
"Asymptotic Distributions of Slope-of-Greatest-Convex-Minorant Estimators." Ann. Statist. 10 (1) 287 - 296, March, 1982. https://doi.org/10.1214/aos/1176345711