asymptotic confidence sequences

todo

Typically, confidence sequences are non-asymptotic objects. There’s no “limit” in the statements as there is with, say, confidence intervals that are constructed via the CLT.

Non-asymptotic results are useful, but typically require stronger assumptions. Eg sub-$\psi$ tail bounds (see sub-psi process) or known bounded moments. But asymptotic results usually require weaker assumptions. Eg Wald interval’s based on the CLT only requires finite variance.

In fact, Bahadur and Savage show that if all you have is finite variance (without a known bound on this variance) then you cannot do non-asymptotic inference.

This suggests the question of whether there exists some notion of asymptotic confidence sequences, which require weaker assumptions to obtain coverage guarantees. Asymptotic confidence sequences were introduced by Waudby-Smith et al.

Definition

Definition

A $(1-\alpha$)-asymptotic CS (asympCS) for a parameter $\mu$ is a sequence of sets $(C_t)$ obeying

$$\underset{k\to \infty }{\lim }\mathbb{P}(m\ge k:\mu \in C_m).$$