testing group invariance

A particular setup in hypothesis testing which generalizes testing for exchangeability, testing for sphericity (which can be used to test the Gaussian-error assumption in linear regression), permutation testing, and sign-flipping tests.

We have some compact topological group $\mathcal{G}$, which is a set of invertible transformations $G:\mathcal{X}\to \mathcal{X}$ which is closed under composition. We observe some data $X\in \mathcal{X}$ drawn from distribution.

We say $X$ is $\mathcal{G}$-invariant if $X\stackrel{d}{=}GX$ for all $G\in \mathcal{G}$. The null and alternative are:

$$H_0:X\text{ is group invariant},H_1:X\text{ is not group invariant}.$$

In particular applications of interest, the group $\mathcal{G}$ is usually huge (eg permutation tests), and it’s infeasible to test each element $G\in \mathcal{G}$. Typically a random subset of $\mathcal{G}$ is chosen.

• Lardy and Perez-Ortiz study sequential tests of group invariance.
• Koning and Hemerik use a deterministic subgroup of $\mathcal{G}$ instead of a random sample.