Blind Minimax Estimation

TL;DR

This paper introduces blind minimax estimators for linear regression that adaptively estimate the parameter set from measurements, outperforming traditional least-squares and Stein's estimators without prior knowledge.

Contribution

The paper proposes a novel blind minimax estimation framework that does not require prior assumptions and extends Stein's estimator to colored noise scenarios.

Findings

BMEs outperform least-squares estimators in mean-squared error

Stein's estimator and positive-part correction are special cases of BMEs

Simulations confirm superior performance of BMEs over previous methods

Abstract

We consider the linear regression problem of estimating an unknown, deterministic parameter vector based on measurements corrupted by colored Gaussian noise. We present and analyze blind minimax estimators (BMEs), which consist of a bounded parameter set minimax estimator, whose parameter set is itself estimated from measurements. Thus, one does not require any prior assumption or knowledge, and the proposed estimator can be applied to any linear regression problem. We demonstrate analytically that the BMEs strictly dominate the least-squares estimator, i.e., they achieve lower mean-squared error for any value of the parameter vector. Both Stein's estimator and its positive-part correction can be derived within the blind minimax framework. Furthermore, our approach can be readily extended to a wider class of estimation problems than Stein's estimator, which is defined only for white noise and non-transformed measurements. We show through simulations that the BMEs generally outperform previous extensions of Stein's technique.