Asymptotics for P-Value--Based Threshold Estimation in Dose-Response and Regression Settings

• Activity Number: 173
• Type: Contributed
• Date/Time: Monday, July 30, 2012 : 10:30 AM to 12:20 PM
• Sponsor: Section on Nonparametric Statistics
• Author(s): Atul Mallik*+ and Moulinath Banerjee and Bodhisattva Sen
• Companies: University of Michigan and University of Michigan and Columbia University
• Keywords: threshold ; change-point ; weighted Poisson process ; empirical process ; stump function ; nonparametric estimation

Abstract

We investigate the limiting behavior of a p-value based procedure for estimating the threshold level at which a regression function takes off from its baseline value- a problem arising in dose-response studies, environmental statistics, engineering and other related fields. We study the procedure under two sampling settings; one where several responses can be obtained at covariate levels and the other involving limited number of response values per covariate. The estimate, constructed using p-values that test for deviation of the regression function from its baseline level, behaves in markedly different manner under the two sampling settings but exhibits similar convergence rates. The main difference lies in the asymptotic distribution which changes from the argmin of a generalized Poisson process to the argmin of an integrated and transformed Gaussian process across the two settings. The smoothness of the regression function in the vicinity of the threshold, characterized by the order of the cusp, decides the rate of convergence. We construct approximate confidence intervals for a few motivating applications and study the coverage performance through simulations.