Wald interval

The Wald interval is a nonasymptotic confidence interval for the mean based on the CLT:

$$C_n=\left\{{\overline{X}}_n\pm z_{1-\alpha /2}{\hat{\sigma }}_n/\sqrt{n}\right\},$$

where ${\hat{\sigma }}_n$ is the empirical variance. Its cousin in hypothesis testing is the Wald test.

An error-bound on the Wald interval can be written in terms of the KS distance:

$$\mathbb{P}(\mu \notin C_n)\le \alpha +2{\rho }_{\text{KS}}(n^{1/2}({\hat{\mu }}_n-\mu )\Vert N(0,1)).$$