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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 manywell-known estimators, such as the regularized M-estimators forhigh-dimensional problems. In this paper, we will first show through theexample of sparse Gaussian sequence model, that the theoretical results underthe classical minimax framework are insufficient for explaining empiricalobservations. In particular, both hard and soft thresholding estimators are(asymptotically) minimax, however, in practice they often exhibit sub-optimalperformances at various signal-to-noise ratio (SNR) levels. The firstcontribution of this paper is to demonstrate that this issue can be resolved ifthe signal-to-noise ratio is taken into account in the construction of theparameter space. We call the resulting minimax framework the signal-to-noiseratio aware minimaxity. The second contribution of this paper is to showcasehow one can use higher-order asymptotics to obtain accurate approximations ofthe SNR-aware minimax risk and discover minimax estimators. The theoreticalfindings obtained from this refined minimax framework provide new insights andpractical guidance for the estimation of sparse signals.

Signal-to-noise ratio aware minimaxity and higher-order asymptotics