A study on the least square estimator of multiple isotonic regression function

Abstract

Consider the problem of pointwise estimation of f in a multiple isotonic regression model Z = f(X1, ... ,Xd) + ε , where Z is the response variable, f is an unknown non-parametric regression function, which is isotonic with respect to each component, and is the error term. In this article, we investigate the behaviour of the least square estimator of f and establish its asymptotic properties. We generalize the greatest convex minorant characterization of isotonic regression estimator for the multivariate case and use it to establish the asymptotic distribution of properly normalized version of the estimator. Moreover, we test whether the multiple isotonic regression function at a fixed point is larger (or smaller) than a specified value or not based on this estimator, and the consistency of the test is established. The practicability of the estimator and the test are shown on simulated and real data as well.