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A pre-test like estimator dominating the least-squares method

Authors
Journal
Journal of Statistical Planning and Inference
0378-3758
Publisher
Elsevier
Publication Date
Volume
138
Issue
10
Identifiers
DOI: 10.1016/j.jspi.2007.12.002
Keywords
  • Pre-Test Estimators
  • Dominating Estimators
  • Regression Analysis
  • Biased Estimation
  • Mean-Squared Error Criterion

Abstract

Abstract We develop a pre-test type estimator of a deterministic parameter vector β in a linear Gaussian regression model. In contrast to conventional pre-test strategies, that do not dominate the least-squares (LS) method in terms of mean-squared error (MSE), our technique is shown to dominate LS when the effective dimension is greater than or equal to 4. Our estimator is based on a simple and intuitive approach in which we first determine the linear minimum MSE (MMSE) estimate that minimizes the MSE. Since the unknown vector β is deterministic, the MSE, and consequently the MMSE solution, will depend in general on β and therefore cannot be implemented. Instead, we propose applying the linear MMSE strategy with the LS substituted for the true value of β to obtain a new estimate. We then use the current estimate in conjunction with the linear MMSE solution to generate another estimate and continue iterating until convergence. As we show, the limit is a pre-test type method which is zero when the norm of the data is small, and is otherwise a non-linear shrinkage of LS.

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