test-martingale

A test martingale for a distribution $P$ is a nonnegative $P$-martingale with initial value 1. That is, it is a stochastic process $(M_t{)}_{t\ge 1}$ with $M_t\ge 0$ for all $t$, $M_1=1$ and

$${\mathbb{E}}_P[M_t|{\mathcal{F}}_{t-1}]\le M_{t-1},$$

where $({\mathcal{F}}_t)$ 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 $1/\alpha$. This constitutes a level-$\alpha$ 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 $\mathcal{P}$ is a family of test martingales $(M^P{)}_{P\in \mathcal{P}}$.

Another useful classification: $M_t$ is a $P$-martingale iff

$$M_t/M_{t-1}=(1+\varphi (X_t)),$$

where $\varphi$ 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.