Some properties of a class of biased regression estimators |
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| Authors: | JM Lowerre |
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| Institution: | Department of Mathematics, Clarkson College, Potsdam, New York 13676, U.S.A. |
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| Abstract: | For the linear statistical model y = Xb + e, X of full column rank estimates of b of the form (C + X′X)+X′y are studied, where C commutes with X′X and Q+ is the Moore-Penrose inverse of Q. Such estimators may have smaller mean square error, component by component than does the least squares estimator. It is shown that this class of estimators is equivalent to two apparently different classes considered by other authors. It is also shown that there is no C such that (C + X′X)+X′Y = My, in which My has the smallest mean square error, component by component. Two criteria, other than tmse, are suggested for selecting C. Each leads to an estimator independent of the unknown b and σ2. Subsequently, comparisons are made between estimators in which the C matrices are functions of a parameter k. Finally, it is shown for the no intercept model that standardizing, using a biased estimate for the transformed parameter vector, and retransforming to the original units yields an estimator with larger tmse than the least squares estimator. |
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