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 regularized M-estimators for high-dimensional problems. In this paper, we first show through the example of a sparse Gaussian sequence model that some of the theoretical results in the classical minimax framework are insufficient to refine empirical observations. In particular, both hard and soft threshold estimators are (asymptotically) minimax, but in practice often exhibit suboptimal performance at various signal-to-noise ratio (SNR) levels. The first contribution of this paper is to demonstrate that this problem can be solved if the signal-to-noise ratio is considered in the construction of the parameter space. The resulting minimax framework is called signal-to-noise-aware minimax. His second contribution in this White his paper describes how to use higher-order asymptotics to obtain an exact approximation of the minimax risk considering his SNR and discover a minimax estimator. to introduce. Theoretical findings gained from this sophisticated minimax framework provide new insights and practical guidance for the estimation of sparse His signals.
