studies $k > K$. Our results bound the MSE difference between the oracle ensembles and two stacking procedures, with and without data reuse. We use these asymptotic results to describe similarities between stacking and multi-study Bayesian hierarchical models when the SPFs are linear. Related bounds have been studied for the single-study setting in [@vdl1] and in the functional aggregation literature [@juditsky2000functional; @juditsky2008learning]. We also illustrate that if the oracle predictions lie within the convex hull of the SPFs limits $(\lim_{n_k} Y_k; k=1,\ldots,K)$, then stacking produces prediction functions that are asymptotically equivalent to the oracle. We finally provide finite sample comparisons of stacking with DR and NDR. We identify a threshold value for the number of datasets $K$, which depends on the cross-study heterogeneity, below which NDR stacking reduces the MSE.

We apply our NDR and DR stacking procedures to predict mortality in Medicare beneficiaries enrolled before 2002. The datasets contain demographic and health-related information of the beneficiaries at the zipcode-level and measurements of air pollutants. We are interested in predicting the number of deaths per 10,000 person-years. In distinct analyses, we partitioned the database into state-specific datasets ($K=50$) and in county-specific dataset ($K=58$). We compare NDR and DR stacking relative 