REGROW e-variable

Stands for “Relative Growth optimality in the worst case.” This is one idea of how to extent the GRO e-variable to composite alternatives. Like GRO, it was coined by Grunwald, De Heide, and Koolen (2023). In practice, it’s considered less often than the GROW e-variable.

Consider a composite alternative; a class of distributions $\mathcal{Q}$. REGROW finds the test supermartingale $(M_t)$ to maximize

$$\underset{Q\in \mathcal{Q}}{\inf }{\mathbb{E}}_Q[\log M_{\tau }-\log M_{\tau }^Q],$$

where $M_{\tau }^Q$ is the GRO process under a particular $Q\in \mathcal{Q}$.

In game-theoretic hypothesis testing (ad hypothesis testing more generally) and $\mathcal{Q}$ is an exponential family being tested against a point null $P$, then the test-martingale obtained from Jeffreys prior is REGROW.