A bound for the Euclidean distance between restricted and unrestricted estimators of parametric functions in the general linear model |
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Authors: | Pawel R Pordzik |
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Institution: | 1.Department of Mathematical and Statistical Methods,Poznan University of Life Sciences,Poznań,Poland |
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Abstract: | Let
^(\varveck)]{\widehat{\varvec{\kappa}}} and
^(\varveck)]r{\widehat{\varvec{\kappa}}_r} denote the best linear unbiased estimators of a given vector of parametric functions
\varveck = \varvecKb{\varvec{\kappa} = \varvec{K\beta}} in the general linear models
M = {\varvecy, \varvecX\varvecb, s2\varvecV}{{\mathcal M} = \{\varvec{y},\, \varvec{X\varvec{\beta}},\, \sigma^2\varvec{V}\}} and
Mr = {\varvecy, \varvecX\varvecb | \varvecR \varvecb = \varvecr, s2\varvecV}{{\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
^(\varveck)]{\widehat{\varvec{\kappa}}} and
^(\varveck)]r{\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 M{{\mathcal M}} and sub-restricted model Mr*{{\mathcal M}_r^*} containing an essential part of the restrictions
\varvecR\varvecb = \varvecr{\varvec{R}\varvec{\beta} = \varvec{r}} with respect to estimating
\varveck{\varvec{\kappa}}. |
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Keywords: | |
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