Doubleestimationfriendly inference for highdimensional misspecified models
Abstract
All models may be wrong  but that is not necessarily a problem for inference. Consider the standard $t$test for the significance of a variable $X$ for predicting response $Y$ whilst controlling for $p$ other covariates $Z$ in a random design linear model. This yields correct asymptotic type~I error control for the null hypothesis that $X$ is conditionally independent of $Y$ given $Z$ under an \emph{arbitrary} regression model of $Y$ on $(X, Z)$, provided that a linear regression model for $X$ on $Z$ holds. An analogous robustness to misspecification, which we term the "doubleestimationfriendly" (DEF) property, also holds for Wald tests in generalised linear models, with some small modifications. In this expository paper we explore this phenomenon, and propose methodology for highdimensional regression settings that respects the DEF property. We advocate specifying (sparse) generalised linear regression models for both $Y$ and the covariate of interest $X$; our framework gives valid inference for the conditional independence null if either of these hold. In the special case where both specifications are linear, our proposal amounts to a small modification of the popular debiased Lasso test. We also investigate constructing confidence intervals for the regression coefficient of $X$ via inverting our tests; these have coverage guarantees even in partially linear models where the contribution of $Z$ to $Y$ can be arbitrary. Numerical experiments demonstrate the effectiveness of the methodology.
 Publication:

arXiv eprints
 Pub Date:
 September 2019
 arXiv:
 arXiv:1909.10828
 Bibcode:
 2019arXiv190910828S
 Keywords:

 Mathematics  Statistics Theory
 EPrint:
 To appear in Statistical Science