Let us do the twist again
Oskar Maria Baksalary, Götz Trenkler, Erkki P. Liski · Statistical Papers · 2013
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.