Section: New Results

Iterative isotone regression

Participant : Arnaud Guyader.

This is a collaboration with Nicolas Hengartner (Los Alamos), Nicolas Jégou (université de Rennes 2) and Eric Matzner–Løber (université de Rennes 2), and with Alexander B. Németh (Babeş Bolyai University) and Sándor Z. Németh (University of Birmingham).

We explore some theoretical aspects of a recent 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 IIR (iterative isotonic regression). The main result in this work [22] states that the estimator is consistent if 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 the von Neumann algorithm in convex analysis. We also analyse how the algorithm can be stopped in practice using a data-splitting procedure.

With the geometrical interpretation linking this iterative method with the von Neumann algorithm, and making a connection with the general property of isotonicity of projection onto convex cones, we derive in [14] another equivalent algorithm and go further in the analysis.