A lower confidence sequence for the changing mean of non-negative right heavy-tailed observations with bounded mean
This work provides a statistical tool for sequential inference in heavy-tailed data, which is incremental but addresses a known bottleneck in handling infinite variance scenarios.
The paper tackles the problem of constructing a non-parametric, non-asymptotic lower confidence sequence for the running average conditional expectation of non-negative right heavy-tailed observations with bounded mean, achieving a slack that converges to zero and outperforming existing methods like the empirical Bernstein supermartingale in finite variance cases.
A confidence sequence (CS) is an anytime-valid sequential inference primitive which produces an adapted sequence of sets for a predictable parameter sequence with a time-uniform coverage guarantee. This work constructs a non-parametric non-asymptotic lower CS for the running average conditional expectation whose slack converges to zero given non-negative right heavy-tailed observations with bounded mean. Specifically, when the variance is finite the approach dominates the empirical Bernstein supermartingale of Howard et. al.; with infinite variance, can adapt to a known or unknown $(1 + δ)$-th moment bound; and can be efficiently approximated using a sublinear number of sufficient statistics. In certain cases this lower CS can be converted into a closed-interval CS whose width converges to zero, e.g., any bounded realization, or post contextual-bandit inference with bounded rewards and unbounded importance weights. A reference implementation and example simulations demonstrate the technique.