Adaptive Threshold Estimation by FDR

This paper addresses the following simple question about sparsity. For the estimation of an $n$-dimensional mean vector $\boldsymbol{\theta}$ in the Gaussian sequence model, is it possible to find an adaptive optimal threshold estimator in a full range of sparsity levels where nonadaptive optimality can be achieved by threshold estimators? We provide an explicit affirmative answer as follows. Under the squared loss, adaptive minimaxity in strong and weak $\ell_p$ balls with $0\le p<2$ is achieved by a class of smooth threshold estimators with the threshold level of the Benjamini-Hochberg FDR rule or its a certain approximation, provided that the minimax risk is between $n^{-\delta_n}$ and $\delta_n n$ for some $\delta_n\to 0$. For $p=0$, this means adaptive minimaxity in $\ell_0$ balls when $1\le \|\boldsymbol{\theta}\|_0\ll n$. The class of smooth threshold estimators includes the soft and firm threshold estimators but not the hard threshold estimator. The adaptive minimaxity in such a wide range is a delicate problem since the same is not true for the FDR hard threshold estimator at certain threshold and nominal FDR levels. The above adaptive minimaxity of the FDR smooth-threshold estimator is established by proving a stronger notion of adaptive ratio optimality for the soft threshold estimator in the sense that the risk for the FDR threshold level is uniformly within an infinitesimal fraction of the risk for the optimal threshold level for each unknown vector, when the minimum risk of nonadaptive soft threshold estimator is between $n^{-\delta_n}$ and $\delta_n n$. It is an interesting consequence of this adaptive ratio optimality that the FDR smooth-threshold estimator outperforms the sample mean in the common mean model $\theta_i=\mu$ when $|\mu|<n^{-1/2}$.