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We develop non-asymptotically justified methods for hypothesis testing about the p-dimensional coefficients in (possibly nonlinear) regression models, where the hypotheses can also be nonlinear in the coefficients. Our (nonasymptotic) control on the Type I and Type II errors holds for fixed n and does not rely on well-behaved estimation error or prediction error; in particular, when the number of restrictions in the null hypothesis is large relative to p-n, we show it is possible to bypass the sparsity assumption on the coefficients (for both Type I and Type II error control), regularization on the estimates of the coefficients, and other inherent challenges in an inverse problem. We also demonstrate an interesting link between our framework and Farkas' lemma (in math programming) under uncertainty, which points to some potential applications of our method outside traditional hypothesis testing.

Concentration Based Inference in High Dimensional Generalized Regression Models (I: Statistical Guarantees)