Monotone Median Regression

Abstract

Suppose that for each real number $t$ in [0, 1] we have a distribution with distribution function $F_t(\bullet)$, mean $\mu(t)$ and median $m(t) (\mu(t)$ and $m(t)$ are referred to as regression functions). Consider the problems of estimating $\mu(\bullet)$ and $m(\bullet)$. In this paper we propose and discuss an estimator, $\hat{m}(\bullet)$, of $m(\bullet)$ which is monotone. This estimator is analogous to the estimator $\hat{\mu}(\bullet)$ of $\mu(\bullet)$ 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 $\hat{m}(\bullet)$ to $m(\bullet)$ are given and a simulation study, where $\hat{m}(\bullet), \hat{\mu}(\bullet)$ and the least squares linear estimator are compared, is discussed.