$M>8$ to $0.0012$ at $M=15$. Here the expectation is taken over $(y_{1,1},\ldots,y_{1,n},y_{2,1},\ldots,y_{2,n})$. This indicates that ${\text{CV}_{\text{ws}}}$-based approach produces a more accurate PF that stacking with data reuse but the advantage decreases if $M$ is large.

In contrast we illustrate that, if we compare ${\text{CV}_{\text{ws}}}$ and stacking with data reuse for generalist predictions, the difference between the resulting estimates of $U(w)$, $\hat U(w;\nu) - \hat U_{\text{ws}}(w;\nu)$, converges to zero faster than the difference between $\hat U(w;\nu)$ and its limit as $n\to\infty$ for any fixed $K$, rendering $\hat U(w;\nu)$ to be asymptotically identical to $\hat U(w;\nu)$.

To see this result, we first consider the example in Section 3.1 with fixed $\mu = (\mu_1,\mu_2)^\intercal$ and bounded $W$. The utility function for stacking with data reuse is $\hat U(w;(1/2,1/2)) = w^\intercal \hat \Sigma w - 2 \hat b^\intercal w + (\overline{y_1^2} + \overline{y_2^2})/2$, where $$\hat \Sigma = \left[
\begin{array}{cc}
\bar y_1^2&\bar y_1\bar y_2\\
\bar y_1\bar y_2&\bar y_2^2
\end{array}
\right],~~ \hat b = \left[
\begin{array}{c}
\bar y_1 \bar y\\
\bar y_2 \bar y
\end{array}
\right],$$ and $\bar y = (\bar y_1 + \bar y_2)/2$. Let $\bar y_{k,-m} = (n(M-1)/M)^{-1}\sum_{i\notin D_{k,m}} y_{k,i}$ and $\bar y_{k,m} = (n/M)^{-1}\sum_{i\in D_{k,m}} y_{k,i}$. We use a 2-fold ${\text{CV}_{\text{ws}}}$ to select 