Title: Relaxing Gaussian assumptions in high dimensional statistical procedures

Abstract: The assumption that high dimensional data is Gaussian is pervasive in many statistical procedures, due not only to its tail decay, but also to the level of analytic tractability this special distribution provides. We explore the relaxation of the Gaussian assumption in Single Index models and Shrinkage estimation using two tools that originate in Stein’s method: Stein kernels, and the zero bias transform. Taking this approach leads to measures of discrepancy from the Gaussian that arise naturally from the nature of the procedures considered, and result in performance bounds in contexts not restricted to the Gaussian. The resulting bounds are tight in the sense that they include an additional term that reflects the cost of deviation from the Gaussian, and vanish for the Gaussian, thus recovering this particular special case.