Joint work with Jonas Striaukas and Mads Stehr, Copenhagen Business School
High-dimensional regression methods typically rely on an exact specification of the underlying model. In many applications, however, the model of interest is only approximately valid, and the approximation error may interact with both the sample size and the dimensionality of the problem.
In this paper, we study estimation and inference in sparse high-dimensional regression models in the presence of such approximation error. We derive non-asymptotic bounds and asymptotic rates for LASSO estimation and develop methods for testing the significance of the high-dimensional regression component. In particular, we propose permutation-based inference and establish conditions under which the resulting tests remain asymptotically valid despite model approximation error. More generally, our results quantify how small the approximation error must be for standard high-dimensional rates and valid inference to be retained.
As a main application, we consider regression models for heavy-tailed data with a covariate-dependent tail index. Under conditional regular variation, observations above a high threshold can be transformed into an approximately linear regression model, where the deviation from an exact Pareto tail appears as an explicit approximation error. This provides a framework for high-dimensional estimation and inference on how covariates affect tail behavior, while allowing the Pareto model to hold only asymptotically.