Asymptotics for <math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si4.gif" display="inline" overflow="scroll" class="math"><mi>p</mi></math>-value based threshold estimation under repeated measurements

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 “repeated measurement” 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 best possible rate of convergence using our method: a “cusp” of order k at the threshold yields a rate of N − 1 / ( 2 k + 1 ) , where N is the size/budget of the problem (total number of responses across all values of covariates). The asymptotic distribution of the normalized estimator is shown to be the minimizer of a generalized compound Poisson process and confidence intervals based on this distribution are proposed. The method is illustrated on simulations and an application to data from a complex queuing system is presented.

我们研究了一种基于p值的程序的大样本行为，该程序用于估计回归函数从其基线值开始变化的阈值水平，这是在剂量 - 反应研究、工程及其他相关领域中出现的一个问题。我们在“重复测量”设置下研究该程序，其中在每个协变量水平上可以获得多个响应。该估计量是通过拟合一个“树桩”函数来近似用于检验回归函数与其基线水平偏差的p值而构建的。阈值附近回归函数的平滑性决定了使用我们的方法所能达到的最佳收敛速度：阈值处k阶“尖点”产生的收敛速度为\(N^{-1/(2k + 1)}\)，其中\(N\)是问题的规模/预算（所有协变量值对应的响应总数）。标准化估计量的渐近分布被证明是一个广义复合泊松过程的极小值点，并基于此分布提出了置信区间。通过模拟对该方法进行了说明，并给出了在一个复杂排队系统数据上的应用。