Admissible anytime-valid sequential inference must rely on nonnegative martingales.
By Aaditya Ramdas, J. Ruf, Martin Larsson, Wouter M. Koolen. 2020
150 citations recorded by the source corpus.
Wald's anytime-valid $p$-values and Robbins' confidence sequences enable sequential inference for composite and nonparametric classes of distributions at arbitrary stopping times, as do more recent proposals involving Vovk's `$e$-values' or Shafer's `betting scores'. Examining the literature, one finds that at the heart of all these (quite different) approaches has been the identification of composite nonnegative (super)martingales. Thus, informally, nonnegative (super)martingales are known to be sufficient for \emph{valid} sequential inference. Our central contribution is to show that martingales are also universal---all \emph{admissible} constructions of (composite) anytime $p$-values, confidence sequences, or $e$-values must necessarily utilize nonnegative martingales (or so-called max-martingales in the case of $p$-values). Sufficient conditions for composite admissibility are also provided. Our proofs utilize a plethora of modern mathematical tools for composite testing and estimation problems: max-martingales, Snell envelopes, and new Doob-Levy martingales make appearances in previously unencountered ways. Informally, if one wishes to perform anytime-valid sequential inference, then any existing approach can be recovered or dominated using martingales. We provide several sophisticated examples, with special focus on the nonparametric problem of testing if a distribution is symmetric, where our new constructions render past methods inadmissible.
Subfields: Statistics and Probability
View Paper: https://www.semanticscholar.org/paper/2a0579d2e3002132ce82469710fa2fbad1a6f21d
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Citations in the corpus
- Estimating means of bounded random variables by betting (2020, methodology): “…M mt , which has at most unit expectation under the null, has been called a betting score [Shafer, 2021] or an e -value [Vovk, 2021] and their relationship to sequential testing [Grünwald et al., 2019] and estimation [Ramdas et al., 2020] as an alternative to p -values has been recently examined.”
- Time-uniform central limit theory and asymptotic confidence sequences (2021, unknown): “[32] for a more general discussion of admissible confidence sequences.”
- False discovery rate control with e‐values (2020, background): “See (Ramdas et al., 2020, 2021) for details.”