Generalized Error Exponents for Small Sample Universal Hypothesis Testing

Huang, Duruo; Meyn, Sean

doi:10.1109/tit.2013.2283266

Cited by 19 publications

(18 citation statements)

References 55 publications

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“…Prior to this work, the question of developing sample-optimal testers in the high-confidence regime has been considered for uniformity testing (and, via Goldreich's reduction, identity testing). Specifically, [HM13] showed that Paninski's uniformity tester (based on the number of unique elements) has the sample-optimal sample complexity of O( n log(1/δ)/ǫ 2 ) in the sublinear sample regime, i.e., when the sample size is o(n). More recently, [DGPP17] gave a different tester that achieves the optimal sample complexity O(( n log(1/δ) + log(1/δ))/ǫ 2 ) in the entire regime of parameters.…”

Section: Prior and Concurrent Workmentioning

confidence: 99%

“…It should be noted that Question 1.1 has received renewed research attention in the information theory and statistics communities. Specifically, [HM13,KBW20] focused on developing testers with improved dependence on δ for uniformity testing [HM13], equivalence and independence testing [KBW20]. Prior to this work, uniformity testing-and, via Goldreich's reduction [Gol16], identity testing-was the only statistical task whose sample complexity had been characterized in the high-confidence regime [DGPP17].…”

Section: Introduction 1background and Motivationmentioning

confidence: 99%

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Optimal Testing of Discrete Distributions with High Probability

Diakonikolas¹,

Gouleakis²,

Kane³

et al. 2020

Preprint

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We study the problem of testing discrete distributions with a focus on the high probability regime. Specifically, given samples from one or more discrete distributions, a property P, and parameters 0 < ǫ, δ < 1, we want to distinguish with probability at least 1 − δ whether these distributions satisfy P or are ǫ-far from P in total variation distance. Most prior work in distribution testing studied the constant confidence case (corresponding to δ = Ω(1)), and provided sample-optimal testers for a range of properties. While one can always boost the confidence probability of any such tester by black-box amplification, this generic boosting method typically leads to sub-optimal sample bounds.Here we study the following broad question: For a given property P, can we characterize the sample complexity of testing P as a function of all relevant problem parameters, including the error probability δ? Prior to this work, uniformity testing was the only statistical task whose sample complexity had been characterized in this setting. As our main results, we provide the first algorithms for closeness and independence testing that are sample-optimal, within constant factors, as a function of all relevant parameters. We also show matching information-theoretic lower bounds on the sample complexity of these problems. Our techniques naturally extend to give optimal testers for related problems. To illustrate the generality of our methods, we give optimal algorithms for testing collections of distributions and testing closeness with unequal sized samples.

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Section: Prior and Concurrent Workmentioning

confidence: 99%

Section: Introduction 1background and Motivationmentioning

confidence: 99%

Optimal Testing of Discrete Distributions with High Probability

Diakonikolas¹,

Gouleakis²,

Kane³

et al. 2020

Preprint

View full text Add to dashboard Cite

show abstract

“…x i j acts as a test statistic to test H 0 , H 1 and H 2 hypotheses. Therefore, with respect to the employed sampling theorem approximation [16], and assuming a narrowband signal over H 1 , x i j can be represented as…”

Section: System Modelmentioning

confidence: 99%

“…• Probability of miss detection, P m , which complements the probability of spectral resource detection is measured by the use of receiver operating characteristics curves as a function of the false alarm probability. It is simply expressed as P m = 1 − P d (16) • Jain Fairness Index, J t , which is used to determine the overall level of fairness in spectral resource allocation among the participating CR vehicular SUs within the secondary network is expressed below:…”

Section: Evaluation Metricsmentioning

confidence: 99%

“…With the application of an ideal BP filter to pre‐filter the received signal, given the observed signal

r_{i j} ()(, t)

, for

0 \leq t \leq T

, the output of the integrator denoted by

x_{i j}

acts as a test statistic to test

H_{0}

H_{1}

and

H_{2}

hypotheses. Therefore, with respect to the employed sampling theorem approximation [16], and assuming a narrowband signal over

H_{1}

x_{i j}

can be represented as

x_{i j} \underline{\underline{def}} \frac{2}{N_{0}} \int_{0}^{T} r_{i j}^{2} (t) normald t

where

N_{0}

denotes the one‐sided noise PSD. Given that

x_{i j}

is a random variable with a chi‐square

()(, χ^{2})

distribution, the test statistics can be expressed as

x_{i J} ≅ \frac{1}{N_{0} W} \{\sum_{i = 1}^{N / 2} [{(α_{c} {double-struckS}_{normalc i} - α_{s} {double-struckS}_{normals i} + W {double-struckG}_{normalc i})}^{2}] + \sum i = 1\}

…”

Section: System Modelmentioning

confidence: 99%

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Cognitive radio‐enabled Internet of Vehicles: a cooperative spectrum sensing and allocation for vehicular communication

et al. 2018

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Internet of Things (IoTs) era is expected to empower all aspects of Intelligent Transportation System (ITS) to improve transport safety and reduce road accidents. US Federal Communication Commission (FCC) officially allocated 75MHz spectrum in the 5.9GHz band to support vehicular communication which many studies have found insufficient. In this paper, we studied the application of Cognitive Radio (CR) technology to IoVs in order to increase the spectrum resource opportunities available for vehicular communication, especially when the officially allocated 75MHz spectrum in 5.9GHz band is not enough due to high demands as a result of increasing number of connected vehicles as already foreseen in the near era of IoTs. We proposed a novel CR Assisted Vehicular NETwork (CRAVNET) framework which empowers CR enabled vehicles to make opportunistic usage of licensed spectrum bands on the highways. We also developed a novel co-operative three-state spectrum sensing and allocation model which makes CR vehicular secondary units (SUs) aware of additional spectrum resources opportunities on their current and future positions and applies optimal sensing node allocation algorithm to guarantee timely acquisition of the available channels within a limited sensing time. The results of the theoretical analyses and simulation experiments have demonstrated that the proposed model can significantly improve the performance of a cooperative spectrum sensing and provide vehicles with additional spectrum opportunities without harmful interference against the Primary Users (PUs) activities.

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Continuous-variable entropic uncertainty relations

Hertz

Cerf

2019

J. Phys. A: Math. Theor.

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Uncertainty relations are central to quantum physics. While they were originally formulated in terms of variances, they have later been successfully expressed with entropies following the advent of Shannon information theory. Here, we review recent results on entropic uncertainty relations involving continuous variables, such as position x and momentum p. This includes the generalization to arbitrary (not necessarily canonically-conjugate) variables as well as entropic uncertainty relations that take x-p correlations into account and admit all Gaussian pure states as minimum uncertainty states. We emphasize that these continuous-variable uncertainty relations can be conveniently reformulated in terms of entropy power, a central quantity in the information-theoretic description of random signals, which makes a bridge with variance-based uncertainty relations. In this review, we take the quantum optics viewpoint and consider uncertainties on the amplitude and phase quadratures of the electromagnetic field, which are isomorphic to x and p, but the formalism applies to all such variables (and linear combinations thereof) regardless of their physical meaning. Then, in the second part of this paper, we move on to new results and introduce a tighter entropic uncertainty relation for two arbitrary vectors of intercommuting continuous variables that takes correlations into account. It is proven conditionally on reasonable assumptions. Finally, we present some conjectures for new entropic uncertainty relations involving more than two continuous variables.

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Generalized Error Exponents for Small Sample Universal Hypothesis Testing

Cited by 19 publications

References 55 publications

Optimal Testing of Discrete Distributions with High Probability

Optimal Testing of Discrete Distributions with High Probability

Cognitive radio‐enabled Internet of Vehicles: a cooperative spectrum sensing and allocation for vehicular communication

Continuous-variable entropic uncertainty relations

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