We are in the setting of game-theoretic hypothesis testing, and now have a composite null. The methods outlined in testing by betting—simple vs simple and testing by betting—simple vs composite outline methods to construct e-values and e-processes which rely heavily on having a point null.
There are two general methods for handling composite nulls:
- universal inference and
- the reverse information projection (RIPr) (or, more generally, the numeraire e-variable).