Inference in Adaptive Regression via the Kac-Rice Formula

We derive an exact p-value for testing a global null hypothesis in a general adaptive regression setting. Our approach uses the Kac-Rice formula (as described in Adler & Taylor 2007) applied to the problem of maximizing a Gaussian process. The resulting test statistic has a known distribution in finite samples, assuming Gaussian errors. We examine this test statistic in the case of the lasso, group lasso, principal components, and matrix completion problems. For the lasso problem, our test relates closely to the recently proposed covariance test of Lockhart et al. (2013). Our approach also yields exact selective inference for the mean parameter at the global maximizer of the process. AMS 2000 subject classifications: Primary 62M40; secondary 62J05.