The asymptotic distribution of the isotonic regression estimator over a general countable pre-ordered set

TL;DR

This paper derives the asymptotic distribution of the isotonic regression estimator over a general countable pre-ordered set, revealing its properties and applications to bimonotone function estimation.

Contribution

It provides the first comprehensive analysis of the limiting distribution of isotonic regression estimators on general pre-ordered sets, including their properties and convergence rates.

Findings

Limiting distribution characterized as concatenation of separate isotonic regressions.

Isotonization preserves the convergence rate of the original estimator.

Applications demonstrated in bimonotone regression and probability mass function estimation.

Abstract

We study the isotonic regression estimator over a general countable pre-ordered set. We obtain the limiting distribution of the estimator and study its properties. It is proved that, under some general assumptions, the limiting distribution of the isotonized estimator is given by the concatenation of the separate isotonic regressions of the certain subvectors of an unrestrecred estimator's asymptotic distribution. Also, we show that the isotonization preserves the rate of convergence of the underlying estimator. We apply these results to the problems of estimation of a bimonotone regression function and estimation of a bimonotone probability mass function.