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Let us do the twist again

Author

Listed:
  • Oskar Baksalary
  • Götz Trenkler
  • Erkki Liski

Abstract

Krämer (Sankhy $$\bar{\mathrm{a }}$$ 42:130–131, 1980 ) posed the following problem: “Which are the $$\mathbf{y}$$ , given $$\mathbf{X}$$ and $$\mathbf{V}$$ , such that OLS and Gauss–Markov are equal?”. In other words, the problem aimed at identifying those vectors $$\mathbf{y}$$ for which the ordinary least squares (OLS) and Gauss–Markov estimates of the parameter vector $$\varvec{\beta }$$ coincide under the general Gauss–Markov model $$\mathbf{y}=\mathbf{X} \varvec{\beta } + \mathbf{u}$$ . The problem was later called a “twist” to Kruskal’s Theorem, which provides conditions necessary and sufficient for the OLS and Gauss–Markov estimates of $$\varvec{\beta }$$ to be equal. The present paper focuses on a similar problem to the one posed by Krämer in the aforementioned paper. However, instead of the estimation of $$\varvec{\beta }$$ , we consider the estimation of the systematic part $$\mathbf{X} \varvec{\beta }$$ , which is a natural consequence of relaxing the assumption that $$\mathbf{X}$$ and $$\mathbf{V}$$ are of full (column) rank made by Krämer. Further results, dealing with the Euclidean distance between the best linear unbiased estimator (BLUE) and the ordinary least squares estimator (OLSE) of $$\mathbf{X} \varvec{\beta }$$ , as well as with an equality between BLUE and OLSE are also provided. The calculations are mostly based on a joint partitioned representation of a pair of orthogonal projectors. Copyright The Author(s) 2013

Suggested Citation

  • Oskar Baksalary & Götz Trenkler & Erkki Liski, 2013. "Let us do the twist again," Statistical Papers, Springer, vol. 54(4), pages 1109-1119, November.
  • Handle: RePEc:spr:stpapr:v:54:y:2013:i:4:p:1109-1119
    DOI: 10.1007/s00362-013-0512-3
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    References listed on IDEAS

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    1. Oskar Baksalary & Götz Trenkler, 2009. "A projector oriented approach to the best linear unbiased estimator," Statistical Papers, Springer, vol. 50(4), pages 721-733, August.
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    Cited by:

    1. Roman Zmyślony & João Mexia & Francisco Carvalho & Inês Sequeira, 2016. "Mean driven balance and uniformly best linear unbiased estimators," Statistical Papers, Springer, vol. 57(1), pages 43-53, March.

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