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least squares \( \theta \) estimator :


\begin{displaymath}
\tilde{\theta }_{L}=\frac{2x+y-z\pm \sqrt{\left( z-y\right) ^{2}+4x^{2}}}{2\left( y-z\right) }\end{displaymath}

where: \( z=\sum ^{m}_{l=1}a_{l}^{2} \), \( x=\sum ^{m}_{l=1}a_{l}b_{l} \) and \( y=\sum ^{m}_{l=1}b_{l}^{2} \) .

to check which of the two solutions for \( \tilde{\theta }_{L} \) provides the minimum, the residual sum of squares, R, should be calculated for each where:


\begin{displaymath}
R=\frac{\left( 2x+y+z\right) \tilde{\theta }_{L}^{2}-2\left(...
...theta }_{L}+z}{1-2\tilde{\theta }_{L}+2\tilde{\theta }_{L}^{2}}\end{displaymath}



Olivier Langella 2002-03-20