Multiple Isotonic Median Regression

We consider the partial order on the unit square; s₁ = (x₁, y₁) s₂ = (x₂, y₂) if and only if xᵢ yᵢ for $i = 1, 2$, and say that a real-valued function f is isotone if s₁ s₂ implies that f(s₁) f(s₂). 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 m̂ and give an algorithm for computing m̂. Furthermore we show that if xᵢⱼ (j = 1, ⋯, nᵢ) are observations at sᵢ (i = 1, ⋯, k) then m̂ minimizes D(f) = ∑ᵏᵢ₌₁ ∑nᵢⱼ₌₁ |f(sᵢ) - xᵢⱼ| 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.