The problem of how one can test the null hypothesis that $P$ has log-concave density when observing a sequence of iid\ data with distribution $P$ over $\mathbb{R}^d$ ask. This paper proves one interesting negative and positive result: the absence of a test (super)martingale and the consistency of universal inference. To elaborate, the set $\mathcal{L}$ of log-concave distributions is a nonparametric class, containing the set $\mathcal G$ of all possible Gaussian distributions with arbitrary mean and covariance. Developing further the recent geometric notion of fork-convexity, we first prove that there is no non-trivial test martingale or test supermartingale for $\mathcal G$ ($\mathcal G processes that are simultaneously non-negative supermartingale for all distributions of $ ). ), and thus its superset $\mathcal{L}$ . Because of this negative result, we turn our attention to building an e-process. It is a process whose expected value is at most 1 at any stopping time under any distribution of $\mathcal{L}$. A \alpha$ test that just thresholds at $1/\alpha$. Adopt a universal reasoning approach. It avoids cumbersome likelihood asymptotics by taking the ratio of the unpredicted likelihood to the alternatives to the maximum likelihood under null. Despite its conservatism, we show that the resulting test is consistent (power 1) and derive its power over Hellinger’s alternative. To our knowledge, there are no other e-process or sequential tests for $\mathcal{L}$.
