Multiple Isotonic Median Regression

Abstract

We consider the partial order on the unit square; $s_1 = (x_1, y_1) \ll s_2 = (x_2, y_2)$ if and only if $x_i \leqq y_i$ for $i = 1, 2$, and say that a real-valued function $f$ is isotone if $s_1 \ll s_2$ implies that $f(s_1) \leqq f(s_2)$. Suppose that for each point, $s$, in the unit square we have a distribution with median $m(s)$ and $m(s)$ is isotone. In this paper we propose an isotone estimator for $m$ which we denote by $\hat{m}$ and give an algorithm for computing $\hat{m}$. Furthermore we show that if $x_{ij} (j = 1, \cdots, n_i)$ are observations at $s_i (i = 1, \cdots, k)$ then $\hat{m}$ minimizes $D(f) = \sum^k_{i=1} \sum^{n_i}_{j=1} |f(s_i) - x_{ij}|$ over all isotone functions $f$. The estimator is also shown to be consistent for $m$ and some rates are given for this convergence. A brief discussion of isotone percentile regression is also given.