## 3 GLS  
It is well known that the GLS method can be applied to \(y_{1}\) and \(y_{2}\) to obtain the best linear unbiased estimate (BLUE) \(\hat{\mu}\) of \(\mu\)[7]. Here, \"best\" means minimum variance and unbiased means that on average (across hypothetical or real realizations of the same experiment), the estimate \(\hat{\mu}\) will equal its true value \(\mu\). The GLS estimate for \(\mu\) arising from the model  
\[\left( \begin{array}{c}y_{1}\\ y_{2}\end{array} \right)=\left( \begin{array}{c}\mu+e_{1}\\ \mu+e_{2}\end{array} \right) \tag{4}\]  
with \(\Sigma=\mathrm{Cov}(e_{1},e_{2})=\mathrm{Cov}(y_{1},y_{2})\) given by  
\[\hat{\mu} = cG^{t}\Sigma^{-1}\left( \begin{array}{c}y_{1}\\ y_{2}\end{array} \right)\]  
where the scalar \(c=(G^{t}\Sigma^{-1}G)^{-1}\). And the variance \(\sigma^{2}\) of the GLS estimate \(\hat{\mu}\) is given by  
\[\sigma^{2} = \left(G^{t}\Sigma^{-1}G\right)^{-1} \tag{6}\]where the matrix \(G\) is given by \(G^{t}=(1,1)\). Putting the covariance of Equation 1 into Equations 5 and 6 gives \(\hat{\mu}=c_{1}y_{1}+c_{2}y_{2}=0.88\) and \(\sigma=0.22\) where \(c_{1}=-0.24\) and \(c_{2}=1.24\). Notice that \(c_{1}+c_{2}=1\) (so that \(\hat{\mu}\) is unbiased) but also that \(c_{1}<0\) and that \(\hat{\mu}\) is smaller than each of the two measured values, \(y_{1}=1.5\) and \(y_{2}=1.0\). GLS is usually introduced in the context of estimating \(\beta\) and future \(y\) values in a linear regression relating the response \(y\) to predictors \(X\) via \(y=X\beta+\epsilon\) ([7]). Therefore, in Equation 4, the mean \(\mu\) plays the role of the unknown \(\beta\). The GLS solution in Equations 5 and 6 then follows from standard calculus or projection matrix results. For example, one can write \(\hat{\mu}=a_{1}y_{1}+(1-a_{1})y_{2}\), note that \(\text{var}(\hat{\mu})=a_{1}^{2}\sigma_{y1}^{2}+(1-a_{1})^{2}\sigma_{y2}^{2}+2 a_{1}(1-a_{1})\sigma_{y1,y2}^{2}\) and solve for \(a_{1}\) to minimize \(\text{var}(\hat{\mu})\) by setting the derivative of \(\text{var}(\hat{\mu})\) with respect to \(a_{1}\) to 0. The GLS solution of Equations 5 and 6 with covariance \(\Sigma\) of Equation 3 is 0.89 \(\pm\) 0.22. Because the last term in Equation 2 is smaller than the others, the \(\Sigma\) in Equation 3 that is slightly modified compared to the \(\Sigma\) in Equation 1 still leads to PPP. But in Section 4 we describe other modifications to \(\Sigma\) that do not lead to PPP. GLS estimation has a long and successful history, but met with serious objection within the nuclear physics community in the context of combining estimates from multiple experiments upon observing a tendency to produce estimates that are outside the range of the data. More specifically, to date the tendency has been to produce estimates that are less than the minimum data value, so have been criticized as being \"too small\" [2]. GLS estimation is guaranteed to produce the BLUE even if the underlying data are not Gaussian. However, if the data is not Gaussian, then the minimum variance unbiased estimator (MVUE) is not necessarily linear in the data. Also, though unbiased estimation might sound politically correct, it is not necessarily superior to biased estimation [8]. Therefore, PPP has motivated the nuclear physics community to consider estimators other than GLS. If the data has a Gaussian distribution, then it is well known that the GLS estimate is the same as the maximum likelihood (ML) estimate. This is because the log of the Gaussian likelihood involves a sum of squares, so choosing an estimate (the GLS estimate) that minimizes a sum of squares corresponds to choosing an estimate (the ML estimate) that maximizes the likelihood. Ordinary LS (OLS), weighted LS (WLS), and GLS are all essentially the same technique, but OLS is used if \(\Sigma\) is proportional to a unit-diagonal matrix, WLS is used if \(\Sigma\) is proportional to a diagonal matrix, and GLS is used if \(\Sigma\) is an arbitrary positive definite covariance matrix.