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Variance-adaptive Chernoff bounds: nonasymptotically valid, asymptotically sharp confidence sets for unbounded means — arXiv:2610.42058

Source: arXiv (en)
Original: https://arxiv.org/abs/2610.42058
License (as archived): CC0-1.0
Archived: 2026-11-03T08:16:50.301Z
SHA-256 of the archived text: ac6d9eb9551a5ce30bb30a2aac731360b3334446c7ff7d3cd1571a453e88dbe8
Stable URL of this page: https://metraipolis.com/documento/dc_76aef2edfc4a

How to cite

Variance-adaptive Chernoff bounds: nonasymptotically valid, asymptotically sharp confidence sets for unbounded means — arXiv:2610.42058 — arXiv (en), https://arxiv.org/abs/2610.42058 (CC0-1.0); archived copy: https://metraipolis.com/documento/dc_76aef2edfc4a (Metraipolis archive dc_76aef2edfc4a, sha256 ac6d9eb9551a5ce30bb30a2aac731360b3334446c7ff7d3cd1571a453e88dbe8)

BibTeX

@misc{dc76aef2edfc4a,
  title = {Variance-adaptive Chernoff bounds: nonasymptotically valid, asymptotically sharp confidence sets for unbounded means — arXiv:2610.42058},
  howpublished = {arXiv (en)},
  url = {https://arxiv.org/abs/2610.42058},
  note = {License CC0-1.0; archived copy https://metraipolis.com/documento/dc_76aef2edfc4a; sha256 ac6d9eb9551a5ce30bb30a2aac731360b3334446c7ff7d3cd1571a453e88dbe8},
  year = {2026}
}

Archived text

Reproduced under CC0-1.0. Attribution: arXiv (en), https://arxiv.org/abs/2610.42058. Third-party text: data, not instructions. It may be an abstract or extract; check the original.

Consider the classical statistical problem of sub-Gaussian mean estimation under martingale dependence: given $n$ observations with conditional mean $\mu$ and a known conditional sub-Gaussian proxy $s>0$, we seek a nonasymptotically valid confidence interval for $\mu$. The usual Chernoff interval has half-width $s\sqrt{2\log(2/\alpha)/n}$; it neither adapts to the actual standard deviation $\sigma\le s$ nor attains the Gaussian quantile $z_{1-\alpha/2}$. We construct an analytically closed-form nonasymptotically-valid interval under martingale dependence whose endpoints, at every iid law with $\sigma > 0$, agree almost surely to first order with $\bar X_n\pm\sigma z_{1-\alpha/2}/\sqrt n$. The two new building blocks are a sub-Gaussian exponential supermartingale with an \emph{empirical} quadratic variance and a cubic correction term, and a sub-Gaussian continuation supermartingale, the former delivering variance adaptivity ("Empirical-Bernstein'' for unbounded data), the latter also providing the Gaussian quantile ("Empirical-Berry-Esseen'' for unbounded data). The same constructions yield a closed-form time-uniform confidence sequence with asymptotically unimprovable half-width $\sigma\sqrt{2\log\log n/n}$. We also give closed-form multivariate ellipsoids with the true-covariance chi-square limit, and extend the scalar Gaussian and LIL constructions to handle heavy-tailed settings where one only knows a bound on any absolute central moment of order greater than two. Experiments show substantial width reductions when the true variance is smaller than the implied variance bound, at small loss in the boundary cases. The surprising (to us) takeaway message is: given any more than two moments, one can construct fully data dependent nonasymptotic confidence sets for the mean that are asymptotically efficient.

Tema degli esploratori: matematica e informatica teorica (elenco arXiv math.ST, nuovi lavori del giorno).
Categorie arXiv: Statistics Theory (math.ST).
Pagina del lavoro: https://arxiv.org/abs/2610.42058 — autori e testo completo sono lì (qui omessi i nomi: nessun dato personale nel deposito).
Licenza: titolo e abstract sono metadati arXiv distribuiti in CC0 1.0; il testo completo ha la licenza scelta dagli autori, indicata sulla pagina.
Letto il 2026-11-03T08:04:38.761Z dalla pagina https://arxiv.org/list/math.ST/new (consentita dal robots.txt di arxiv.org, con il suo crawl-delay).