Sequential Kernelized Independence Testing

Aleksandr, Podkopaev,; Blöbaum, Patrick; Kasiviswanathan, Shiva Prasad; Ramdas, Aaditya

doi:10.48550/arxiv.2212.07383

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“…Test supermartingales are so named because they form the canonical path to sequentially testing composite hypotheses, which is encapsulated entirely by the above relation, in that valid tests can be derived by rejecting only when a test supermartingale crosses a threshold. They are particularly interesting in nonparametric settings; for example one can use them to sequentially test the mean of a bounded random variable [WR23], for testing symmetry [RRLK20], for two-sample testing [SR21], independence testing [PBKR22], and testing calibration [AHZ21], to mention only a few interesting sequential nonparametric problems. We shall discuss test supermartingales extensively in this paper.…”

Section: Problem Setup and Backgroundmentioning

confidence: 99%

A Sequential Test for Log-Concavity

Gangrade¹,

Rinaldo²,

Ramdas³

2023

Preprint

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On observing a sequence of i.i.d. data with distribution P on R d , 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 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 G (a process that is simultaneously a nonnegative supermartingale for every distribution in G), and hence also for its superset L. Due to this negative result, we turn our attention to constructing an eprocess -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.

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