Baseline zone estimation in two dimensions

TL;DR

This paper introduces a method for estimating convex baseline regions in two-dimensional non-parametric regression, analyzing convergence rates under different sampling schemes and considering extensions to non-convex regions.

Contribution

It proposes a novel convex contour estimation algorithm based on fitting a stump function to p-value maps, with theoretical convergence analysis in two sampling settings.

Findings

Convergence rate of $N^{2/(4p+3)}$ in dose-response setting

Convergence rate of $N^{1/(2p+2)}$ in standard regression

Effective extension to non-convex baseline regions

Abstract

We consider the problem of estimating the region on which a non-parametric regression function is at its baseline level in two dimensions. The baseline level typically corresponds to the minimum/maximum of the function and estimating such regions or their complements is pertinent to several problems arising in edge estimation, environmental statistics, fMRI and related fields. We assume the baseline region to be convex and estimate it via fitting a `stump' function to approximate $p$-values obtained from tests for deviation of the regression function from its baseline level. The estimates, obtained using an algorithm originally developed for constructing convex contours of a density, are studied in two different sampling settings, one where several responses can be obtained at a number of different covariate-levels (dose-response) and the other involving limited number of response values per covariate (standard regression). The shape of the baseline region and the smoothness of the regression function at its boundary play a critical role in determining the rate of convergence of our estimate: for a regression function which is `p-regular' at the boundary of the convex baseline region, our estimate converges at a rate $N^{2/(4p+3)}$ in the dose-response setting, $N$ being the total budget, and its analogue in the standard regression setting converges at a rate of $N^{1/(2p+2)}$. Extensions to non-convex baseline regions are explored as well.