Cluster-Seeking James–Stein Estimators

This paper considers the problem of estimating a high-dimensional vector of parameters Θ ∈ Rn from a noisy observation. The noise vector is independent identically distributed Gaussian with known variance. For a squared-error loss function, the James-Stein (JS) estimator is known to dominate the simple maximum-likelihood (ML) estimator when the dimension n exceeds two. The JS-estimator shrinks the observed vector toward the origin, and the risk reduction over the ML-estimator is greatest for Θ that lie close to the origin. JS-estimators can be generalized to shrink the data toward any target subspace. Such estimators also dominate the ML-estimator, but the risk reduction is significant only when Θ lies close to the subspace. This leads to the question: in the absence of prior information about Θ, how do we design estimators that give significant risk reduction over the ML-estimator for a wide range of Θ? In this paper, we propose shrinkage estimators that attempt to infer the structure of Θ from the observed data in order to construct a good attracting subspace. In particular, the components of the observed vector are separated into clusters, and the elements in each cluster shrunk toward a common attractor. The number of clusters and the attractor for each cluster are determined from the observed vector. We provide concentration results for the squared-error loss and convergence results for the risk of the proposed estimators. The results show that the estimators give significant risk reduction over the ML-estimator for a wide range of Θ, particularly for large n. Simulation results are provided to support the theoretical claims.