A test martingale for a distribution $P$ is a nonnegative $P$-martingale with initial value 1. That is, it is a stochastic process $(M_{t})_{t≥1}$ with $M_{t}≥0$ for all $t$, $M_{1}=1$ and

$E_{P}[M_{t}∣F_{t−1}]≤M_{t−1},$where $(F_{t})$ is the filtration to which $(M_{t})$ is adapted.

Test martingales are common tools in game-theoretic probability and game-theoretic statistics where they represent fair bets under $P$. Often one generalizes the discussion of test martingales to e-process, as they are bounded above by test martingales and thus also obey Ville’s inequality. Both e-processes and test-martingales thus immediately yield sequential hypothesis testing for the null $H_{0}:P=P_{0}$: we reject when the process exceeds $1/α$. This constitutes a level-$α$ sequential test.

Note that while test-martingales and e-processes are useful for defining sequential tests, they can also be considered bonafide measures of evidence on their own. Eg the larger a $P$-test-martingale grows over time, the more evidence this is against $P$. Thus they can be used not only for the Neyman-Pearson paradigm but also satisfy Fisher’s paradigm.

Often we are interested in statistical problems involving sets of distributions, e.g., testing composite nulls. For such problems, it’s useful to have a family of martingales. A test martingale *family* for a set of distributions $P$ is a family of test martingales $(M_{P})_{P∈P}$.

Another useful classification: $M_{t}$ is a $P$-martingale iff

$M_{t}/M_{t−1}=(1+ϕ(X_{t})),$where $ϕ$ is a bounded odd-function.

Interestingly, sometimes test-martingales will appear in reduced filtrations and not in the original filtration. See coarsened filtrations can increase power and testing exchangeability.