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 m and give an algorithm for computing m. Furthermore we show that if x₈₉ (j = 1, , nᵢ) are observations at sᵢ (i = 1, , k) then m minimizes D (f) = ᵏ₈=₁ ^nᵢ₉=₁ |f (sᵢ) - x₈₉| 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.