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 non-asymptotic 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 non-asymptotic 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，针对凸回归函数极小点与极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下展开研究，其中过程的性能通过单个函数来评估。提出了完全自适应且计算高效的算法，并给出了极小点与极小值在估计精度和置信区间期望长度方面的尖锐极小极大下界。非渐近局部极小极大框架揭示了极小点与极小值联合估计与推断中的新现象。我们建立了一个新颖的不确定性原理，该原理为任意凸回归函数下同时估计极小点与极小值的能力提供了基本极限。关于极小点与极小值的置信区间期望长度，也存在类似的结论。