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Optimality of the least squares estimator

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  • Berk, Robert
  • Hwang, Jiunn T.

Abstract

In a standard linear model, we explore the optimality of the least squares estimator under assuptions stronger than those for the Gauss-Markov theorem. The conclusion is then much stronger than that of the Gauss-Markov theorem. Specifically, two results are cited below: Under the assumption that the unobserved error [var epsilon] has a spherically symmetric distribution, the least squares estimator for the regression coefficient [beta] is shown to maximize the probability that [beta] - [beta] stays in any symmetric convex set among linear unbiased estimators [beta]. With the additional assumption that [var epsilon] is unimodal, the conclusion holds among equivariant estimators. The import of these results for risk functions is also discussed.

Suggested Citation

  • Berk, Robert & Hwang, Jiunn T., 1989. "Optimality of the least squares estimator," Journal of Multivariate Analysis, Elsevier, vol. 30(2), pages 245-254, August.
    • Handle: RePEc:eee:jmvana:v:30:y:1989:i:2:p:245-254
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    Cited by:

    1. Albisetti, Isaia & Balabdaoui, Fadoua & Holzmann, Hajo, 2020. "Testing for spherical and elliptical symmetry," Journal of Multivariate Analysis, Elsevier, vol. 180(C).
    2. Kariya, Takeaki & Kurata, Hiroshi, 2002. "A Maximal Extension of the Gauss-Markov Theorem and Its Nonlinear Version," Journal of Multivariate Analysis, Elsevier, vol. 83(1), pages 37-55, October.
    3. Gneiting, Tilmann, 1998. "On[alpha]-Symmetric Multivariate Characteristic Functions," Journal of Multivariate Analysis, Elsevier, vol. 64(2), pages 131-147, February.

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