|
Abstract:
|
Stein's formula states that a random variable of the form z?f(z)?div[f](z) is mean-zero for functions f with integrable gradient. Here, div[f] is the divergence of the function f and z is a standard normal vector. This paper aims to propose a Second Order Stein formula to characterize the variance of such random variables for all functions f(z) with square integrable gradient, and to demonstrate the usefulness of this formula in various applications. In the Gaussian sequence model, a consequence of Stein's formula is Stein's Unbiased Risk Estimate (SURE), an unbiased estimate of the mean squared risk for almost any estimator of the unknown mean. A first application of the Second Order Stein formula is an Unbiased Risk Estimate for SURE itself (SURE for SURE): an unbiased estimate providing information about the squared distance between SURE and its target. SURE for SURE has a simple form as a function of the data and is applicable to all ?^ with square integrable gradient, e.g. the Lasso and the Elastic Net. In addition to SURE for SURE, time permitting, other applications in high-dimensional statistics will be presented.
|
