How small is negligible? Rethinking equivalence testing with e-values

A classical statistical problem is to assess whether an unknown quantity is negligible: practically equivalent to zero. It is standard to define 'negligible' as being smaller in magnitude than some threshold margin. Specifying this margin has plagued statisticians for decades: if it is set too large then one can hardly speak of negligibility, but if the margin is set too small then one may need an enormous amount of data to statistically establish negligibility. We study this problem in depth and show how e-values can be used to bypass it, by enabling one to select the margin post-hoc: after seeing the data. As an illustration in machine learning, we turn the sequential fairness audits of Chugg et al. (2023) into anytime-valid certificates of fairness that require no prespecified tolerance.