Finite Sample Confidence Regions for Linear Regression Parameters Using Arbitrary Predictors
This provides a robust framework for statistical inference in linear models, though it appears incremental as it builds on existing confidence region methods with added flexibility.
The paper tackles the problem of constructing confidence regions for linear regression parameters using arbitrary predictors, with minimal noise assumptions and flexibility for near-linear functions, and validates the method on synthetic data.
We explore a novel methodology for constructing confidence regions for parameters of linear models, using predictions from any arbitrary predictor. Our framework requires minimal assumptions on the noise and can be extended to functions deviating from strict linearity up to some adjustable threshold, thereby accommodating a comprehensive and pragmatically relevant set of functions. The derived confidence regions can be cast as constraints within a Mixed Integer Linear Programming framework, enabling optimisation of linear objectives. This representation enables robust optimization and the extraction of confidence intervals for specific parameter coordinates. Unlike previous methods, the confidence region can be empty, which can be used for hypothesis testing. Finally, we validate the empirical applicability of our method on synthetic data.