Asymptotics for p-value based threshold estimation in dose-response settings

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 arising in dose-response studies, engineering and other related fields. We study the procedure under the so called “dose-response” setting, where several responses can be obtained at each covariate-level. The estimator is constructed via fitting a “stump” function to approximate p-values that test for deviation of the regression function from its baseline level. The smoothness of the regression function in the vicinity of the threshold determines the optimal rate of convergence: a “cusp” of order k at the threshold yields an optimal rate of N−1/(2k+1), where N is the total budget. The asymptotic distribution of the normalized estimator is shown to be the minimizer of a generalized compound Poisson process. A limited simulation study is used to illustrate the method and an application to data from a complex queuing system presented.