A two-stage hybrid procedure for estimating an inverse regression function

TL;DR

This paper introduces a two-stage hybrid method combining isotonic regression and local linear approximation to efficiently estimate inverse regression functions, achieving parametric convergence rates and improving estimation of derivatives.

Contribution

The paper proposes a novel two-stage hybrid procedure and a bootstrapped variant that enhance inverse regression estimation accuracy and convergence speed.

Findings

The second-stage estimator attains the parametric $n^{1/2}$ convergence rate.

The bootstrapped variant (BTSP) is consistent and overcomes slow convergence issues.

Simulation studies demonstrate the effectiveness of the proposed methods.

Abstract

We consider a two-stage procedure (TSP) for estimating an inverse regression function at a given point, where isotonic regression is used at stage one to obtain an initial estimate and a local linear approximation in the vicinity of this estimate is used at stage two. We establish that the convergence rate of the second-stage estimate can attain the parametric $n^{1/2}$ rate. Furthermore, a bootstrapped variant of TSP (BTSP) is introduced and its consistency properties studied. This variant manages to overcome the slow speed of the convergence in distribution and the estimation of the derivative of the regression function at the unknown target quantity. Finally, the finite sample performance of BTSP is studied through simulations and the method is illustrated on a data set.