Aggregating many estimators using estimated weights

Consider an increasing number of consistent estimators to be averaged when only estimated weights are available. The underlying parameter of interest can be identical across estimators (homogeneity) or not (heterogeneity). The contribution of the paper is threefold. First, it is shown that the interaction of the estimated weights with the estimators can generate specific bias terms. This constrains the number of estimators that can be aggregated when weight estimation is ignored in inference. Second, the paper proposes estimated adaptive weights, which allow for standard Gaussian inference in a uniform manner and are asymptotically optimal both under homogeneity and heterogeneity. Third, conditions ensuring the validity of the Cochran (1937) Q test of homogeneity with estimated variance are given.