## 3 Estimation error in \(\hat{\Sigma}\)  
In nearly all real situations, \(\Sigma\) is not known exactly, but must be estimated. Suppose the estimate \(\hat{\Sigma}=S\), where \(S\) is the sample covariance matrix defined as (\(t\) denotes transpose) based on \(n\) pairs of \(y_{1},y_{2}\) values that are auxiliary data pairs separate from the \(y_{1}=1.5\) and \(y_{2}=1.0\) values. We will consider two examples.  
* Example 3.1 Assume that the distribution of \((n-1)S\) is Wishart with \(\
u=n-1\) degrees of freedom, which is exactly true if the \(y_{1}\) and \(y_{2}\) pairs are bivariate normal, and approximately true for many other bivariate distributions [18]. Then the root mean squared error (RMSE = \(\sqrt{E(\hat{\mu}-\mu)^{2}}\), where \(E\) denotes expected value) in estimating the true mean \(\mu\) using GLS, is 0.46, 0.32, 0.27, 0.25, 0.23, and 0.22 for \(\
u=2,3,4,5,10,\text{ and }30\), respectively. These RMSE estimates are based on 10,000 simulations in _R_[19] and are repeatable to \(\pm 0.01\). And the RMSE is 0.27 using the simple mean of \(y_{1}\) and \(y_{2}\). Recall that if \(\Sigma\) is known exactly, then the RMSE in \(\hat{\mu}=0.22\). Therefore, if \(n\leq 5\), the naive sample mean has the same or lower RMSE than does GLS. A third estimation option suggested in [20] uses only the variances of \(\hat{\Sigma}\), not because \(y_{1}\) and \(y_{2}\) are thought to be independent, but because the estimated covariance \(\hat{\sigma}_{12}^{2}\) is thought to be unreliable. For \(\
u\,=\,2,3,4,5,10,\text{ and }30\), option 3 has an RMSE of 0.25, which is the theoretical best RMSE corresponding to the situation in which \(\sigma_{1}\) and \(\sigma_{2}\) are known exactly. Therefore, there is merit in [PERSON]'s [20] suggested option 3 in this case, because it has lower RMSE for all values of \(\
u\) than the simple mean and lower RMSE than GLS for small \(\
u\) values. Whether option 1 (GLS), option 2 (naive sample mean), or option 3 (use variances only, not covariance) is preferred depends on the true data likelihood and on the true \(\Sigma\).
* Example 3.2 Still assuming a normal likelihood but changing \(\sigma_{1}^{2},\sigma_{2}^{2},\text{ and }\sigma_{12}^{2}\) in \(\Sigma\) to 0.1134, 0.02, and 0.0475, respectively (cor(\(y_{1},y_{2}\)) = 0.997), options 1, 2, and 3 have RMSE of 0.025, 0.24, and 0.17, respectively, for \(\
u=3\), and 0.04, 0.24, and 0.17, respectively, for \(\
u=2\). This \(\Sigma\) corresponds to a very strong PPP effect with high correlation between \(y_{1}\) and \(y_{2}\), so even with large estimation error in \(\hat{\Sigma}\), the GLS method has by far the lowest RMSE for all values of \(\
u\). We note that as \(\
u\rightarrow\infty\), the GL \(\hat{\mu}\) has \(\sigma_{\hat{\mu}}\to 0.017\). As another example, let \(\sigma_{1}^{2},\sigma_{2}^{2},\text{ and }\sigma_{12}^{2}\) in \(\Sigma\) be 0.1134, 0.0104, and 0.02, respectively. Then options 1, 2, and 3 have RMSE of 0.13, 0.20, and 0.12, respectively, for \(\
u=3\), and RMSE of 0.25, 0.20, and 0.13, respectively, for \(\
u=2\). So in this case option 3 is again a good choice for small values of \(\
u\). We note that as \(\
u\rightarrow\infty\), the GLS \(\hat{\mu}\) has \(\sigma_{\hat{\mu}}\to 0.096\).