Non-asymptotic confidence region construction in metric spaces

Recent advancements in non-asymptotic inference have significantly impacted modern statistics and machine learning. In this paper, we utilize sub-Gaussian and sub-exponential concentration inequalities to quantify the uncertainty of random elements within general metric spaces. Specifically, utilizing these inequalities, we construct non-asymptotic confidence regions for the unbounded, asymmetric, and independent centered random vectors in Hilbert spaces. An improved symmetrization inequality guarantees tighter upper bounds for these vectors. Moreover, we establish non-asymptotic confidence regions for the unbounded, independent centered random vectors in general metric spaces. Furthermore, we establish finite sample theory under mild and finite moment conditions and create model-free confidence regions using robust median-of-mean estimators. Both simulated and empirical studies show that the proposed confidence regions significantly outperform those based on the sample mean.