w_2) = (\nu_1,\nu_2)$, while the oracle weights $w_g = (w_{g,1}, w_{g,2})$ that optimize $U_g(w)$ are $$w_g = \left\{
\begin{array}{cc}
\left(\frac{|\bar y_2|}{|\bar y_1| + |\bar y_2|}, \frac{|\bar y_1|}{|\bar y_1| + |\bar y_2|}\right) & \bar y_1\cdot \bar y_2 < 0, \\
(1,0)     & |\bar y_1|\leq |\bar y_2|,~~\bar y_1\cdot \bar y_2 \geq 0,\\
(0,1)     & |\bar y_1|> |\bar y_2|,~~\bar y_1\cdot \bar y_2 \geq 0.
\end{array}\right.$$

The oracle weights favor $\hat Y_1$, i.e. $w_{g,1}>w_{g,2}$, if $\text{cMSE}(\hat Y_1) < \text{cMSE}(\hat Y_2)$, where cMSE indicates the conditional MSE of a SPF $\hat Y_{t}^\ell$: $$\text{cMSE}{(\hat Y)} = \int_{(x,y)} \left(y - \hat Y(x)\right)^2 dP_0(x,y).$$ The cMSE measures the actual prediction performance of $\hat Y_t^\ell$ across studies given the observed data $(y_{1,1},\ldots,y_{1,n},y_{2,1},\ldots,y_{2,n})$. Note that in our example, $\text{cMSE}(\hat Y_k) = |\bar y_k|^2 + 2$, $k=1,2$. On the other hand, $\hat w$ favor $\hat Y_k$ whenever $\nu_k>\nu_{k'}$, regardless of the cMSE of each SPF.

To understand the discrepancy described above, we examine the bias of $\hat U(w;K^{-1}{\mathbf{1}}_K)$ to $U_g(w)$, defined as $\mathbb E\left(\hat U(w;K^{-1}{\mathbf{1}}_K) - U_g(w)\right)$, where the expectation is taken over all observed data $(y_{1,1},\ldots,y_{1,n},y_{2,1},\ldots,y_{2,n})$. By definition, we have $$\begin{aligned}
\hat U(w;K^{-1}{\mathbf{1}}_K) - U_g(w) =& \underbrace{2(\nu_1w_2 + \nu_2w_1)\bar y_1\bar y_2+ 2(\nu_1w_1 \bar y_1^2 + \nu_2w_2 \bar y_2^2)}_{\text{data-reuse}}\\
+& 2 