Adaptive estimation of high-dimensional signal-to-noise ratios

摘要: We consider the equivalent problems of estimating residual variance, proportion explained variance $\eta$ and signal strength in a high-dimensional linear regression model with Gaussian random design. Our aim is to understand impact not knowing sparsity vector coefficients distribution design on minimax estimation rates $\eta$. Depending $k$ coefficients, optimal estimators either rely or are based $U$-type statistics. In important situation where unknown, we build an adaptive procedure whose convergence rate simultaneously achieves risk over all up logarithmic loss which prove be non avoidable. Finally, knowledge shown play critical role. When consistent indeed possible much narrower regimes than for known distribution.