Monotone Regression Estimates for Grouped Observations

Abstract

The maximum likelihood estimator of a nondecreasing regression function with normally distributed errors has been considered in the literature. Its asymptotic distribution at a point is related to a solution of the heat equation, and its rate of convergence to the underlying regression function is of order $n^{-1/3}$. This estimator can be modified by grouping adjacent observations and then "isotonizing" the corresponding means. It is shown that the resulting estimator has an asymptotic normal distribution for certain group sizes and its rate of convergence is of order $n^{-2/5}$. The results of a simulation study for small sample sizes are presented and grouping procedures are discussed.