Estimation and inference for minimizer and minimum of convex functions: Optimality, adaptivity and uncertainty principles
在白噪声和非参数回归模型下，研究了凸回归函数最小值和最小点的最优估计与推断，提出了完全自适应且计算高效的算法，并给出了估计精度和置信区间期望长度的尖锐极小化下界，同时揭示了同时估计与推断的不确定性原理。
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.