We propose an e-value based framework for testing arbitrary composite nulls against composite alternatives, when an ϵ fraction of the data can be arbitrarily corrupted. Our tests are inherently sequential, being valid at arbitrary data-dependent stopping times, but they are new even for fixed sample sizes, giving type-I error control without any regularity conditions. We first prove that least favourable distribution (LFD) pairs, when they exist, yield optimal e-values for testing arbitrary composite nulls against composite alternatives. Then we show that if an LFD pair exists for some composite null and alternative, then the LFDs of Huber’s ϵ-contamination or total variation (TV) neighborhoods around that specific pair form the optimal LFD pair for the corresponding robustified composite hypotheses. Furthermore, where LFDs do not exist, we develop new robust composite tests for general settings. Our test statistics are a nonnegative supermartingale under the (robust) null, even under a sequentially adaptive (non-i.i.d.) contamination model where the conditional distribution of each observation given the past data lies within an ϵ TV ball of some distribution in the original composite null. When LFDs exist, our supermartingale grows to infinity exponentially fast under any distribution in the (ϵ TV-corruption of the) alternative at the optimal rate. When LFDs do not exist, we provide an asymptotic growth rate analysis, showing that as ϵ → 0, the exponent converges to the corresponding Kullback–Leibler divergence, recovering the classical optimal non-robust rate. Simulations validate the theory and demonstrate reasonable practical performance.

In this paper, we address the problem of testing exchangeability of a sequence of random variables, $X_1, X_2,\cdots$. This problem has been studied under the recently popular framework of \emph{testing by betting}. But the mapping of testing problems to game is not one to one: many games can be designed for the same test. Past work established that it is futile to play single game betting on every observation: test martingales in the data filtration are powerless. Two avenues have been explored to circumvent this impossibility: betting in a reduced filtration (wealth is a test martingale in a coarsened filtration), or playing many games in parallel (wealth is an e-process in the data filtration). The former has proved to be difficult to theoretically analyze, while the latter only works for binary or discrete observation spaces. Here, we introduce a different approach that circumvents both drawbacks. We design a new (yet simple) game in which we observe the data sequence in pairs. Even though betting on individual observations is futile, we show that betting on pairs of observations is not. To elaborate, we prove that our game leads to a nontrivial test martingale, which is interesting because it has been obtained by shrinking the filtration very slightly. We show that our test controls type-1 error despite continuous monitoring, and is consistent for both binary and continuous observations, under a broad class of alternatives. Due to the shrunk filtration, optional stopping is only allowed at even stopping times: a relatively minor price. We provide a variety of simulations that align with our theoretical findings.