Abstract
Let \({\widehat{\varvec{\kappa}}}\) and \({\widehat{\varvec{\kappa}}_r}\) denote the best linear unbiased estimators of a given vector of parametric functions \({\varvec{\kappa} = \varvec{K\beta}}\) in the general linear models \({{\mathcal M} = \{\varvec{y},\, \varvec{X\varvec{\beta}},\, \sigma^2\varvec{V}\}}\) and \({{\mathcal M}_r = \{\varvec{y},\, \varvec{X}\varvec{\beta} \mid \varvec{R} \varvec{\beta} = \varvec{r},\, \sigma^2\varvec{V}\}}\), respectively. A bound for the Euclidean distance between \({\widehat{\varvec{\kappa}}}\) and \({\widehat{\varvec{\kappa}}_r}\) is expressed by the spectral distance between the dispersion matrices of the two estimators, and the difference between sums of squared errors evaluated in the model \({{\mathcal M}}\) and sub-restricted model \({{\mathcal M}_r^*}\) containing an essential part of the restrictions \({\varvec{R}\varvec{\beta} = \varvec{r}}\) with respect to estimating \({\varvec{\kappa}}\).
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Open Access This is an open access article distributed under the terms of the Creative Commons Attribution Noncommercial License (https://creativecommons.org/licenses/by-nc/2.0), which permits any noncommercial use, distribution, and reproduction in any medium, provided the original author(s) and source are credited.
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Pordzik, P.R. A bound for the Euclidean distance between restricted and unrestricted estimators of parametric functions in the general linear model. Stat Papers 53, 299–304 (2012). https://doi.org/10.1007/s00362-010-0336-3
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DOI: https://doi.org/10.1007/s00362-010-0336-3