Monotone Median Regression

Suppose that for each real number t in [0, 1] we have a distribution with distribution function Fₜ(•), mean μ(t) and median m(t) (μ(t) and $m(t)$ are referred to as regression functions). Consider the problems of estimating μ(•) and m(•). In this paper we propose and discuss an estimator, m̂(•), of m(•) which is monotone. This estimator is analogous to the estimator μ̂(•) of μ(•) which was explored by Brunk (1970) (Estimation of isotonic regression in Nonparametric Techniques in Statistical Inference, Cambridge University Press, 177-195). Rates for the convergence of m̂(•) to m(•) are given and a simulation study, where m̂(•), μ̂(•) and the least squares linear estimator are compared, is discussed.