Optimal estimation and inference for both the minimizer and minimum of a convex regression function under the white noise and nonparametric regression models are studied in a nonasymptotic local minimax framework, where the performance of a procedure is evaluated at individual functions. Fully adaptive and computationally efficient algorithms are proposed and sharp minimax lower bounds are given for both the estimation accuracy and expected length of confidence intervals for the minimizer and minimum. The nonasymptotic local minimax framework brings out new phenomena in simultaneous estimation and inference for the minimizer and minimum. We establish a novel uncertainty principle that provides a fundamental limit on how well the minimizer and minimum can be estimated simultaneously for any convex regression function. A similar result holds for the expected length of the confidence intervals for the minimizer and minimum.

翻译：在白噪声和非参数回归模型下，针对凸回归函数的最小值点和最小值的最优估计与推断问题，本文在非渐近局部极小极大框架中进行了研究，其中方法的性能在单个函数上进行评估。提出了完全自适应且计算高效的算法，并给出了最小值点和最小值的估计精度及置信区间期望长度的尖锐极小极大下界。非渐近局部极小极大框架揭示了最小值点和最小值联合估计与推断中的新现象。我们建立了一个新的不确定性原理，该原理给出了对于任意凸回归函数，无法同时精确估计最小值点和最小值的基本极限。类似结论也适用于最小值点和最小值的置信区间期望长度。