Mean squared error and selection in multivariate calibration

Consider multivariate linear calibration of a single standard. We show that a selection of the q' most informative of the q responses gives a finite mean squared error for the generalised least squares prediction of an unknown standard provided 1 [less-than-or-equals, slant] q' [less-than-or-equals, slant] q and q [greater-or-equal, slanted] 3. Furthermore, the prediction mean is finite provided 1 [less-than-or-equals, slant] q' [less-than-or-equals, slant] q and q [greater-or-equal, slanted] 2 and in general the rth absolute moment exists for 1 [less-than-or-equals, slant] q' [less-than-or-equals, slant] q [greater-or-equal, slanted] r + 1. These results extend those of Lieftinck-Koeijers (1988) to the selection of responses and also sharpen her results when selection is not present, since she showed only that the mean is finite for q [greater-or-equal, slanted] 3 and the mean squared error finite for q [greater-or-equal, slanted] 5.