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Abstract:
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We develop Second Order Stein formulas for statistical inference with high-dimensional data. In the simplest form, the Second Order Stein formula characterizes the variance of the difference between the divergence of a vector-valued function of a standard Gaussian vector and the inner product of the vector and the function. A first application of the Second Order Stein formula is an Unbiased Risk Estimate for Stein's Unbiased Risk Estimator (SURE for SURE). SURE for SURE has a simple form and can be computed explicitly for almost differentiable estimators, for example the Lasso and the Elastic Net. SURE for SURE can be used to select tuning parameters for regularized estimators such as the Lasso. Other applications of the Second Order Stein formula are discussed. This is joint work with Pierre Bellec and Chong Wu.
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