## 2 Background  
A situation that leads to PPP is as follows. Two experiments aim to estimate a physical constant \(\mu\). Experiment one produces a measurement \(y_{1}\), and experiment two produces \(y_{2}\). Measurements \(y_{1}\) and \(y_{2}\) arise by imperfect conversion of underlying measurements \(m_{1}\) and \(m_{2}\). Suppose \(m_{1}=\mu+\epsilon_{R1}\) and \(m_{2}=\mu+\epsilon_{R2}\) where \(\epsilon_{R1}\) is zero-mean random error in \(m_{1}\) and similarly for \(\epsilon_{R2}\). [PERSON] _et al_. [2] prove that if \(y_{1}=m_{1}+\epsilon_{S}\) and \(y_{2}=m_{2}+\alpha\epsilon_{S}\), where \(\epsilon_{S}\sim N(0,\sigma_{S})\) and \(\alpha\) is any positive scale factor other than 1, the covariance matrix \(\Sigma\) of \((y_{1},y_{2})\) can lie in the PPP region in which the GLS estimate of \(\mu\) lies outside the range of \(y_{1}\) and \(y_{2}\). And, [PERSON] _et al_. [2] give an example for which this error model is appropriate. Let \(\Sigma\) be the 2-by-2 symmetric covariance matrix for \(y_{1}\) and \(y_{2}\) with diagonal entries \(\sigma_{1}^{2}\), \(\sigma_{2}^{2}\), and off-diagonal entry \(\sigma_{12}\), which denote the variance of \(y_{1}\), the variance of \(y_{2}\), and the covariance of \(y_{1}\) and \(y_{2}\), respectively. For the case considered here and in [PERSON] and [PERSON] [4]  
\[\Sigma = \left( \begin{array}{cc}0.1134&0.06\\ 0.06&0.0504\end{array} \right) \tag{1}\]  
and we assume throughout that \(\Sigma\) exactly equals the numerical values given in Equation 1.  
If the three errors \(\epsilon_{R1},\epsilon_{R2},\mbox{and}\ \epsilon_{S}\) have normal (Gaussian) distributions, then the joint pdf of \((y_{1},y_{2})\) is also normal. If the three errors have non-normal distributions, then the joint pdf of \((y_{1},y_{2})\) will not be normal and might be difficult to calculate analytically. Therefore, to handle non-normal distributions that might be difficult to calculate analytically, Sections 4 and 5 develop an estimator based on ABC. This ABC-based estimation can be applied with the error model just described resulting from the two experiments producing \(y_{1},y_{2}\) values with a covariance given by Equation 1, and can accommodate non-normal distributions. It is well known that GLS applied to \(y_{1}\) and \(y_{2}\) results in the best linear unbiased estimate (BLUE) \(\hat{\mu}\) of \(\mu\)[16]. 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 \(\mu\). GLS properties are well known and are reviewed in [2]. The GLS estimate for \(\mu\) arising from the generic model  
\[\left( \begin{array}{c}y_{1}\\ y_{2}\end{array} \right)=\left( \begin{array}{c}\mu+e_{1}\\ \mu+e_{2}\end{array} \right) \tag{2}\]  
with \(\Sigma=\mathrm{Cov}(e_{1},e_{2})=\mathrm{Cov}(y_{1},y_{2})\) is given by  
\[\hat{\mu} = cG^{t}\Sigma^{-1}\left( \begin{array}{c}y_{1}\\ y_{2}\end{array} \right) \tag{3}\]  
where the scalar \(c=(G^{t}\Sigma^{-1}G)^{-1}\). And the variance \(\sigma^{2}\) of the GLS estimate \(\hat{\mu}\) is given by  
\[\sigma^{2} = \left(G^{t}\Sigma^{-1}G\right)^{-1} \tag{4}\]  
where the sensitivity (or design) matrix \(G\) is given by \(G^{t}=(1,1)\). Putting the covariance of Equation 1 into Equations 3 and 4 and following the example in [PERSON] _et al._[2] with \(y_{1}=1.5\) and \(y_{2}=1.0\) gives \(\hat{\mu}=c_{1}y_{1}+c_{2}y_{2}=0.89\) and \(\sigma=0.22\) where \(c_{1}=-0.22\) and \(c_{2}=1.22\). 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. GLS estimation is guaranteed to produce the BLUE even if the underlying data are not normal. However, if the data are not normal, then the minimum variance unbiased estimator is not necessarily linear in the data. Also, unbiased estimation is not necessarily superior to biased estimation [17]. The main alternative to GLS estimation in this context is estimation that uses the pdf.