E-values for exponential families: The general case

We analyze common types of e-variables and e-processes for composite exponential family nulls: the optimal e-variable based on the reverse information projection (RIPr), a conditional (COND) e-variable, and the universal inference (UI) and sequentialized RIPr e-processes. Whereas earlier derivations of the RIPr e-variable, for parametric and nonparametric nulls alike, were restricted to cases in which it reduces to a simple-vs.-simple likelihood, we manage to derive it also in ‘anti-simple’ cases in which it cannot be so reduced. We characterize the RIPr for simple and Bayes-mixture based alternatives, either precisely (for Gaussian nulls and alternatives) or in an approximate sense (general exponential family nulls). We also provide conditions under which the RIPr e-variable is (again exactly vs. asymptotically) equal to the COND e-variable, and we determine, up to o(1), the e-power of the four e-statistics as a function of sample size. For d-dimensional null and alternative, the e-power of UI tends to be smaller by a term of (d∕2)logn+O(1) than that of the COND e-variable, which is the clear winner.