Threshold estimation based on a P -value framework

We use p–values as a discrepancy criterion for identifying the threshold value at which a regression function takes off from its baseline value – a problem that is motivated by applications in dose-response studies, astronomy and engineering. In this paper, we study the problem in a controlled sampling setting, where multiple responses, discrete or continuous, can be obtained at a number of different covariate-levels. Our procedure involves testing the hypothesis that the regression function is at its baseline at each covariate value using the sampled responses at that value and then computing the p–value of the test. An estimate of the threshold is provided by fitting a stump, i.e., a piecewise constant function with a single jump discontinuity, to the observed p–values, since the corresponding p–values behave in markedly different ways on different sides of the threshold. The estimate is shown to be consistent, as both the number of covariate values and the number of responses sampled at each value become large, and its finite sample properties are studied through an extensive simulation study. Our approach is computationally simple and can also be used to estimate the baseline value of the regression function. The procedure is illustrated on a motivating real data application.