Weighted Linear Least Squares Estimation WLS

From Eq. (6.14) it suggests itself that squared variance weighting wyi 1 /s2 should be applied. In analytical practice, frequently relative weights 1 

As a rule the calibration coefficients do not alter significantly when WLS is applied instead of OLS. On the other hand, the uncertainty in the lower range of calibration is reduced in a remarkable way. The estimate of the residual standard deviation is 

Other quantities characterizing uncertainties can be estimated in analogy to Eqs. (6.20)-(6.33). Today, software packages for regression analysis usually allow one to enter an estimate of the functional dependence sy = J (y ) and to carry out a suitable weighting with this or other functions offered. 

In general, the decision on weighted or unweighted least squares can be reached on the basis of a statistical test, (see Eq. 6.2.5), or on the basis of a theoretical model. 

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