In non-asymptotic statistical inferences, the variance-type parameters of sub-Gaussian distributions are of paramount importance. However, directly estimating these parameters using the empirical moment generating function is infeasible. To address this, we suggest using the sub-Gaussian intrinsic moment norm achieved by maximizing a sequence of normalized moments. In practice, we offer an intuitive method for checking sub-Gaussian data with a finite sample size using the sub-Gaussian plot. Intrinsic moment norm can be robustly estimated via a simple plug-in approach. Our theoretical findings are also applicable to the multi-armed bandit scenario.

In robust statistical learning, we thoroughly study a large family of robust statistical regressions by the proposed log-truncated M-estimator under the condition that the data have (1+ε)-th moment. With the Lipschitz conditions on the given loss functions and few moment assumptions, we obtain the excess risk bound and the consistency of various convex regressions [such as robust quantile regression and robust GLMs] as well as non-convex regressions including robust deep neural network regressions.

The first part is joint work with Prof. Guang Cheng, and Dr Haoyu Wei. The second part is joint work with Prof. Fang Yao and Lihu Xu, and Dr Qiuran Yao.