Double-Estimation-Friendly Inference for High-Dimensional Misspecified Models

doi:10.1214/22-STS850

2023-02-01

Rajen D. Shah *

Peter Bühlmann

DOI：10.1214/22-STS850

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摘要

En 中文

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 while controlling for p other covariates Z in a random design linear model. This yields correct asymptotic type I error con-trol for the null hypothesis that X is conditionally independent of Y given Z under an arbitrary regression model of Y on (X, Z), provided that a linear regression model for X on Z holds. An analogous robustness to misspecifi-cation, which we term the double-estimation-friendly (DEF) property, also holds for Wald tests in generalised linear models, with some small modifica-tions.In this expository paper, we explore this phenomenon, and propose methodology for high-dimensional regression settings that respects the DEF property. We advocate specifying (sparse) generalised linear regression mod-els 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 investi-gate 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 experi-ments demonstrate the effectiveness of the methodology.

Keyword:

Conditional independence

high-dimensional in-ference

Debiased Lasso

generalised linear models

double robustness