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Let us do the twist again. (English) Zbl 1416.62386
Summary: W. Kraemer [Sankhyā, Ser. A 42, 130–131 (1980; Zbl 0485.62062)] posed the following problem: “Which are the $$y$$, given $$X$$ and $$V$$, such that OLS and Gauss-Markov are equal?”. In other words, the problem aimed at identifying those vectors $$y$$ for which the ordinary least squares (OLS) and Gauss-Markov estimates of the parameter vector $$X\beta$$ coincide under the general Gauss-Markov model $$y=X\beta +u$$. 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 $$\beta$$ 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 $$\beta$$, we consider the estimation of the systematic part $$X\beta$$, which is a natural consequence of relaxing the assumption that $$X$$ and $$V$$ 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 $$X\beta$$, 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.

##### MSC:
 62J05 Linear regression; mixed models 62H12 Estimation in multivariate analysis 15A24 Matrix equations and identities
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##### References:
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