Approximation, characterization, and continuity of multivariate monotonic regression functions

We deal with monotonic regression of multivariate functions [Formula: see text] on a compact rectangular domain [Formula: see text] in [Formula: see text], where monotonicity is understood in a generalized sense: as isotonicity in some coordinate directions and antitonicity in some other coordinate directions. As usual, the monotonic regression of a given function [Formula: see text] is the monotonic function [Formula: see text] that has the smallest (weighted) mean-squared distance from [Formula: see text]. We establish a simple general approach to compute monotonic regression functions: namely, we show that the monotonic regression [Formula: see text] of a given function [Formula: see text] can be approximated arbitrarily well — with simple bounds on the approximation error in both the [Formula: see text]-norm and the [Formula: see text]-norm — by the monotonic regression [Formula: see text] of grid-constant functions [Formula: see text]. monotonic regression algorithms. We also establish the continuity of the monotonic regression [Formula: see text] of a continuous function [Formula: see text] along with an explicit averaging formula for [Formula: see text]. And finally, we deal with generalized monotonic regression where the mean-squared distance from standard monotonic regression is replaced by more complex distance measures which arise, for instance, in maximum smoothed likelihood estimation. We will see that the solution of such generalized monotonic regression problems is simply given by the standard monotonic regression [Formula: see text].