In sequential hypothesis testing based on e-values (i.e., game-theoretic hypothesis testing), one wants to find e-values and e-processes which grow quickly under the alternative. “Quickly” is a non-technical condition, however. What exactly should be maximized? Various authors have used different conditions based on the particular problem at hand.
See:
- GRO e-variable for simple alternatives, equivalent to maximizing log-wealth.
- GROW e-variable for composite alternatives.
- REGROW e-variable for composite alternatives
Often, we do not have a particular alternative that we are considering (eg when building confidence sequences via estimating means by betting, or in many financial applications). In that case, pursuing explicit GRO-like strategies may not make sense. Still, we want the wealth to grow when the null is not true so we still study betting strategies in this case. They are often motivated by the conditions above but not necessarily identical to any of them.