  This article introduces a new nonparametric method for estimating a
univariate regression function of bounded variation. The method exploits the
Jordan decomposition which states that a function of bounded variation can be
decomposed as the sum of a non-decreasing function and a non-increasing
function. This suggests combining the backfitting algorithm for estimating
additive functions with isotonic regression for estimating monotone functions.
The resulting iterative algorithm is called Iterative Isotonic Regression
(I.I.R.). The main technical result in this paper is the consistency of the
proposed estimator when the number of iterations $k_n$ grows appropriately with
the sample size $n$. The proof requires two auxiliary results that are of
interest in and by themselves: firstly, we generalize the well-known
consistency property of isotonic regression to the framework of a non-monotone
regression function, and secondly, we relate the backfitting algorithm to Von
Neumann's algorithm in convex analysis.
