A general framework for heterogeneous variable importance: Pointwise and uniform inference

In complex modeling that may lack explicit parametric structures, understanding the contribution of specific covariates to explaining or predicting a response variable is essential. In particular, it is crucial to examine whether this contribution varies with certain characteristic variables, such as age in psychological studies. We refer to this varying contribution as "heterogeneous variable importance," a concept that allows for evaluating variable relevance across different groups of individuals. To quantify heterogeneous variable importance, we introduce a measure defined as the ratio of two conditional mean squared errors. We then propose a point estimator for this ratio parameter and establish its pointwise and uniform convergence rates. Additionally, we develop procedures for constructing asymptotic confidence intervals and bands, which are guaranteed to achieve their nominal coverage rates. Moreover, the proposed approach demonstrates satisfactory finite-sample performance in simulation studies, and is further illustrated through its application to a real data set.