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Since its development, the minimax framework has been one of the cornerstones of theoretical statistics, and has contributed to the popularity of many well-known estimators, such as the regularized M-estimators for high-dimensional problems. In this paper, we will first show through the example of sparse Gaussian sequence model, that the theoretical results under the classical minimax framework are insufficient for explaining empirical observations. In particular, both hard and soft thresholding estimators are (asymptotically) minimax, however, in practice they often exhibit sub-optimal performances at various signal-to-noise ratio (SNR) levels. The first contribution of this paper is to demonstrate that this issue can be resolved if the signal-to-noise ratio is taken into account in the construction of the parameter space. We call the resulting minimax framework the signal-to-noise ratio aware minimaxity. The second contribution of this paper is to showcase how one can use higher-order asymptotics to obtain accurate approximations of the SNR-aware minimax risk and discover minimax estimators. The theoretical findings obtained from this refined minimax framework provide new insights and practical guidance for the estimation of sparse signals. 

Abstract Precise tuning of chemical reactions with predictable and controllable manners, an ultimate goal chemists desire to achieve, is valuable in the scientific community. This tunability is necessary to understand and regulate chemical transformations at both macroscopic and single-molecule levels to meet demands in potential application scenarios. Herein, we realise accurate tuning of a single-molecule Mizoroki-Heck reaction via applying gate voltages as well as complete deciphering of its detailed intrinsic mechanism by employing an in-situ electrical single-molecule detection, which possesses the capability of single-event tracking. The Mizoroki-Heck reaction can be regulated in different dimensions with a constant catalyst molecule, including the molecular orbital gating of Pd(0) catalyst, the on/off switching of the Mizoroki-Heck reaction, the promotion of its turnover frequency, and the regulation of each elementary reaction within the Mizoroki-Heck catalytic cycle. These results extend the tuning scope of chemical reactions from the macroscopic view to the single-molecule approach, inspiring new insights into designing different strategies or devices to unveil reaction mechanisms and discover novel phenomena. 

Precise time trajectories and detailed reaction pathways of the Diels-Alder reaction were directly observed using accurate single-molecule detection on an in situ label-free single-molecule electrical detection platform. This study demonstrates the well-accepted concerted mechanism and clarifies the role of charge transfer complexes with endo or exo configurations on the reaction path. An unprecedented stepwise pathway was verified at high temperatures in a high-voltage electric field. Experiments and theoretical results revealed an electric field–catalyzed mechanism that shows the presence of a zwitterionic intermediate with one bond formation and variation of concerted and stepwise reactions by the strength of the electric field, thus establishing a previously unidentified approach for mechanistic control by electric field catalysis.