2023/01/09 by Aditya Gangrade, Alessandro Rinaldo, Gangrade, Aditya +3 · 2 citations
Computer Science · Mathematics · #Bayesian Methods and Mixture Models #FOS: Computer and information sciences #FOS: Mathematics #Markov Chains and Monte Carlo Methods #Methodology (stat.ME) #Statistical Methods and Inference #Statistics Theory (math.ST)
paper · pdf · doi:10.48550/arxiv.2301.03542
openalex publication_date 2023/01/09 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
On observing a sequence of i.i.d. data with distribution P on ℝd, we ask the question of how one can test the null hypothesis that P has a log-concave density. This paper proves one interesting negative and positive result: the non-existence of test (super)martingales, and the consistency of universal inference. To elaborate, the set of log-concave distributions L is a nonparametric class, which contains the set \mathcal G of all possible Gaussians with any mean and covariance. Developing further the recent geometric concept of fork-convexity, we first prove that there do no exist any nontrivial test martingales or test supermartingales for \mathcal G (a process that is simultaneously a nonnegative supermartingale for every distribution in \mathcal G), and hence also for its superset L. Due to this negative result, we turn our attention to constructing an e-process -- a process whose expectation at any stopping time is at most one, under any distribution in L -- which yields a level-α test by simply thresholding at 1/α. We take the approach of universal inference, which avoids intractable likelihood asymptotics by taking the ratio of a nonanticipating likelihood over alternatives against the maximum likelihood under the null. Despite its conservatism, we show that the resulting test is consistent (power one), and derive its power against Hellinger alternatives. To the best of our knowledge, there is no other e-process or sequential test for L.