On the sub-Gaussianity of the Beta and Dirichlet distributions

Marchal, Olivier; Arbel, Julyan

doi:10.1214/17-ecp92

Cited by 43 publications

(33 citation statements)

References 13 publications

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“…)} by the central limit theorem and delta method. Furthermore, we give a bound on the tail probability of B • on the basis of the sub-Gaussianity of the Beta distribution (Marchal and Arbel, 2017). Since both B • and 1−B…”

Section: Proof Of Theoremmentioning

confidence: 99%

Interval Estimation for Operating Characteristic Of Continuous Biomarkers With Controlled Sensitivity or Specificity

Huang¹,

Parakati²,

Patil³

et al. 2023

STAT SINICA

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The receiver operating characteristic (ROC) curve provides a comprehensive performance assessment of a continuous biomarker over the full threshold spectrum. Nevertheless, a medical test often dictates to operate at a certain high level of sensitivity or specificity. A diagnostic accuracy metric directly targeting the clinical utility is specificity at the controlled sensitivity level, or vice versa. While the empirical point estimation is readily adopted in practice, the nonparametric interval estimation is challenged by the fact that the variance involves density functions due to estimated threshold. In addition, even with a fixed threshold, many standard confidence intervals including the Wald interval for binomial proportion could have erratic behaviors. In this article, we are motivated by the superior performance of the score interval for binomial proportion and propose a novel extension for the biomarker problem. Meanwhile,

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Section: Proof Of Theoremmentioning

confidence: 99%

Interval Estimation for Operating Characteristic Of Continuous Biomarkers With Controlled Sensitivity or Specificity

Huang¹,

Parakati²,

Patil³

et al. 2023

STAT SINICA

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“…Since the beta distribution is the conjugate prior of Bernoulli, binomial, geometric, and negative binomial, it is often used as the prior distribution for proportions in Bayesian inference. Recently, Marchal and Arbel (2017) proved that the beta distribution is

((), \frac{1}{4 false(α + β + 1 false)})

‐sub‐Gaussian, in the sense that the moment generating function of Z ∼ Beta( α , β ) satisfies

𝔼 ([], \exp ((), t ((), Z - \frac{α}{α + β}))) \leq \exp ((), \frac{t^{2}}{8 false(α + β + 1 false)})

for all

t \in ℝ

. They further gave an upper bound on tail probability:

ℙ ((), Z \geq \frac{α}{α + β} + x) \lor ℙ ((), Z \leq \frac{α}{α + β} - x) \leq \exp ((), - 2 false(α + β + 1 false) x^{2}), \forall x \geq 0 .

…”

Section: Sharp Tail Bounds Of Specific Distributionsmentioning

confidence: 99%

On the non‐asymptotic and sharp lower tail bounds of random variables

Zhang

Zhou

2020

Stat

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The non‐asymptotic tail bounds of random variables play crucial roles in probability, statistics, and machine learning. Despite much success in developing upper bounds on tail probabilities in literature, the lower bounds on tail probabilities are relatively fewer. In this paper, we introduce systematic and user‐friendly schemes for developing non‐asymptotic lower bounds of tail probabilities. In addition, we develop sharp lower tail bounds for the sum of independent sub‐Gaussian and sub‐exponential random variables, which match the classic Hoeffding‐type and Bernstein‐type concentration inequalities, respectively. We also provide non‐asymptotic matching upper and lower tail bounds for a suite of distributions, including gamma, beta, (regular, weighted, and noncentral) chi‐square, binomial, Poisson, Irwin–Hall, etc. We apply the result to establish the matching upper and lower bounds for extreme value expectation of the sum of independent sub‐Gaussian and sub‐exponential random variables. A statistical application of signal identification from sparse heterogeneous mixtures is finally considered.

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“…This fact is known via Theorem 2.1 and Theorem 3.1 of Buldygin and Moskvichova (2013); see also the discussion in the introduction of Marchal and Arbel (2017). Here, we focus on a rather different approach, based on function h and Corollary 2.2, where…”

Section: Bernoulli and Binomial Distributionsmentioning

confidence: 99%

“…The result is known for the beta distribution (Marchal and Arbel, 2017). In this article, we provide proofs for the Bernoulli, binomial, Kumaraswamy and triangular distributions.…”

Section: Introductionmentioning

confidence: 96%

“…This proof technique was used by Marchal and Arbel (2017) for obtaining the optimal proxy variance of the beta and Dirichlet distributions. However we find more convenient to use the conditions stated in Proposition 2.3 using the function h to address the issues presented in this article, except for the triangular distribution in Section 4.2 where this method is employed for a numerical evaluation of σ 2 opt .…”

Section: Introductionmentioning

confidence: 99%

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On strict sub-Gaussianity, optimal proxy variance and symmetry for bounded random variables

2020

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We investigate the sub-Gaussian property for almost surely bounded random variables. If sub-Gaussianity per se is de facto ensured by the bounded support of said random variables, then exciting research avenues remain open. Among these questions is how to characterize the optimal sub-Gaussian proxy variance? Another question is how to characterize strict sub-Gaussianity, defined by a proxy variance equal to the (standard) variance? We address the questions in proposing conditions based on the study of functions variations. A particular focus is given to the relationship between strict sub-Gaussianity and symmetry of the distribution. In particular, we demonstrate that symmetry is neither sufficient nor necessary for strict sub-Gaussianity. In contrast, simple necessary conditions on the one hand, and simple sufficient conditions on the other hand, for strict sub-Gaussianity are provided. These results are illustrated via various applications to a number of bounded random variables, including Bernoulli, beta, binomial, uniform, Kumaraswamy, and triangular distributions.

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On the sub-Gaussianity of the Beta and Dirichlet distributions

Cited by 43 publications

References 13 publications

Interval Estimation for Operating Characteristic Of Continuous Biomarkers With Controlled Sensitivity or Specificity

Interval Estimation for Operating Characteristic Of Continuous Biomarkers With Controlled Sensitivity or Specificity

On the non‐asymptotic and sharp lower tail bounds of random variables

On strict sub-Gaussianity, optimal proxy variance and symmetry for bounded random variables

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