Asymptotics for $p$-value based threshold estimation in regression settings

Abstract

We investigate the large sample 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 that frequently arises in environmental statistics, engineering and other related fields. The estimate is constructed via fitting a “stump” function to approximate $p$-values obtained from tests for deviation of the regression function from its baseline level. The smoothness of the regression function in the vicinity of the threshold determines the rate of convergence: a “cusp” of order $k$ at the threshold yields an optimal convergence rate of $n^{-1/{(2k+1)}}$, $n$ being the number of sampled covariates. We show that the asymptotic distribution of the normalized estimate of the threshold, for both i.i.d. and short range dependent errors, is the minimizer of an integrated and transformed Gaussian process. We study the finite sample behavior of confidence intervals obtained through the asymptotic approximation using simulations, consider extensions to short-range dependent data, and apply our inference procedure to two real data sets.