A bound for the Euclidean distance between restricted and unrestricted estimators of parametric functions in the general linear model
Paweł R. Pordzik · Statistical Papers · 2010
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 $${\varvec{\kappa}}$$ .