Optimal biased estimation using Lehmann-unbiasedness

Abstract

This paper deals with non-Bayesian parameter estimation under the mean-squared-error (MSE), which is a topic of great interest in various engineering fields. Although the unbiasedness condition is commonly used in non-Bayesian MSE estimation, in many cases biased estimation may result in better performance. However, no method for determining the optimal bias function in general cases is available. We propose a new approach for uniform minimum MSE biased estimation, where the optimal bias is chosen in accordance with Lehmann-unbiasedness definition. The proposed approach is based on modifying the MSE risk by its multiplication with a weighting function of the unknown parameter, g <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sup> . Under this modified risk, Lehmann's definition of unbiasedness provides a condition referred to as g-unbiasedness. By using the g-unbiasedness, we derive a novel Cramér-Rao-type lower bound on the MSE of locally g-unbiased estimators. In addition, we show that if there exists an estimator that achieves the new bound, then it is produced by the penalized maximum likelihood estimator with a penalty function log g. Simulations show that the proposed approach can lead to non-trivial estimators with lower MSE than existing mean-unbiased estimators.