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Cluster-Seeking James-Stein Estimators

Accepted version
Peer-reviewed

Type

Article

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Authors

Srinath, K Pavan 
Venkataramanan, Ramji  ORCID logo  https://orcid.org/0000-0001-7915-5432

Abstract

This paper considers the problem of estimating a high-dimensional vector of parameters θRn from a noisy observation. The noise vector is i.i.d. 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 towards 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 towards 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 towards 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.

Description

Keywords

high-dimensional estimation, large deviations bounds, loss function estimates, risk estimates, shrinkage estimators

Journal Title

IEEE Transactions on Information Theory

Conference Name

Journal ISSN

0018-9448
1557-9654

Volume Title

64

Publisher

IEEE
Sponsorship
European Commission (631489)
Isaac Newton Trust (1321(c))
Engineering and Physical Sciences Research Council (EP/N013999/1)
Isaac Newton Trust (1540 (R))
Marie Curie Career Integration Grant; 10.13039/501100004815-Early Career Grant from the Isaac Newton Trust