From multiple Gaussian sequences to functional data and beyond: a Stein estimation approach

We expand the notion of Gaussian sequence models to n experiments and propose a Stein estimation strategy which relies on pooling information across experiments. An oracle inequality is established to assess conditional risks given the underlying effects, based on which we can quantify the size of relative error and obtain a tuning‐free recovery strategy that is easy to compute, produces model parsimony and extends to unknown variance. We show that the simultaneous recovery is adaptive to an oracle strategy, which also enjoys a robustness guarantee in a minimax sense. A connection to functional data is established, via Le Cam theory, for fixed and random designs under general regularity settings. We further extend the model projection to general bases with mild conditions on correlation structure and conclude with potential application to other statistical problems. Simulated and real data examples are provided to lend empirical support to the methodology proposed and to illustrate the potential for substantial computational savings.