A characterization of distributions by mean values of statistics and certain probabilistic metrics

Zinger, A. A.; Kakosyan, Ashot V.; Klebanov, Lev B.

doi:10.1007/bf01099119

Cited by 20 publications

(14 citation statements)

References 5 publications

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“…We elaborate on the arguments laid out in Section 6.4. By Corollary 1 (with v 1 and v 2 defined there), the test with bin size 1 κ has (statistical) size at most v 1 L (a √ log m) 2 , so by the union bound, the combined test (35) has size at most b max v 1 L (a 2 log m) ≤ c 1 a 2 , for a constant c 1 that depends only on (d, L). Now fix an alternative, meaning with f ≠ g both in H s d (L) with such that f − g 2 ≥ ε.…”

Section: Proof Of Propositionmentioning

confidence: 97%

Remember the curse of dimensionality: the case of goodness-of-fit testing in arbitrary dimension

Arias-Castro

Pelletier

Saligrama

2018

Journal of Nonparametric Statistics

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Despite a substantial literature on nonparametric two-sample goodness-of-fit testing in arbitrary dimensions spanning decades, there is no mention there of any curse of dimensionality. Only more recently Ramdas et al. [25] have discussed this issue in the context of kernel methods by showing that their performance degrades with the dimension even when the underlying distributions are isotropic Gaussians. We take a minimax perspective and follow in the footsteps of Ingster [16] to derive the minimax rate in arbitrary dimension when the discrepancy is measured in the L 2 metric. That rate is revealed to be nonparametric and exhibit a prototypical curse of dimensionality. We further extend Ingster's work to show that the chi-squared test achieves the minimax rate. Moreover, we show that the test can be made to work when the distributions have support of low intrinsic dimension. Finally, inspired by Ingster [18], we consider a multiscale version of the chi-square test which can adapt to unknown smoothness and/or unknown intrinsic dimensionality without much loss in power.

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Section: Proof Of Propositionmentioning

confidence: 97%

Remember the curse of dimensionality: the case of goodness-of-fit testing in arbitrary dimension

Arias-Castro

Pelletier

Saligrama

2018

Journal of Nonparametric Statistics

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“…The quantity N ( μ , ν ) can be shown [7] to be a metric in the space of all probability measures R d , so that the null hypothesis in two-sample comparisons can be formulated as H 0 : N ( μ , ν ) = 0. A normalized version of N can be derived as , where…”

Section: Resultsmentioning

confidence: 99%

Multivariate search for differentially expressed gene combinations

et al. 2004

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Background: To identify differentially expressed genes, it is standard practice to test a twosample hypothesis for each gene with a proper adjustment for multiple testing. Such tests are essentially univariate and disregard the multidimensional structure of microarray data. A more general two-sample hypothesis is formulated in terms of the joint distribution of any sub-vector of expression signals.

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“…A proof of the basic energy inequality, D ( F , G ) ≥ 0 with equality if and only if F = G follows from Ref and also from Mattner's result . An alternate proof related to a result of Morgenstern appears in Refs .…”

Section: Background and Application Of Energy Distancementioning

confidence: 95%

“…Generalizations and interesting special cases of the energy distance have appeared in the recent literature; see Refs . A similar idea related to energy distance and E ‐statistics were considered as N ‐distances and N ‐statistics in Ref ; see also Ref . Measures of the energy distance type have also been studied in the machine learning literature …”

Section: Background and Application Of Energy Distancementioning

confidence: 99%

Energy distance

Rizzo

Székely²

2015

WIREs Computational Stats

159

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Energy distance is a metric that measures the distance between the distributions of random vectors. Energy distance is zero if and only if the distributions are identical, thus it characterizes equality of distributions and provides a theoretical foundation for statistical inference and analysis. Energy statistics are functions of distances between observations in metric spaces. As a statistic, energy distance can be applied to measure the difference between a sample and a hypothesized distribution or the difference between two or more samples in arbitrary, not necessarily equal dimensions. The name energy is inspired by the close analogy with Newton's gravitational potential energy. Applications include testing independence by distance covariance, goodness‐of‐fit, nonparametric tests for equality of distributions and extension of analysis of variance, generalizations of clustering algorithms, change point analysis, feature selection, and more. WIREs Comput Stat 2016, 8:27–38. doi: 10.1002/wics.1375 This article is categorized under: Statistical and Graphical Methods of Data Analysis > Multivariate Analysis Statistical and Graphical Methods of Data Analysis > Nonparametric Methods

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A characterization of distributions by mean values of statistics and certain probabilistic metrics

Cited by 20 publications

References 5 publications

Remember the curse of dimensionality: the case of goodness-of-fit testing in arbitrary dimension

Remember the curse of dimensionality: the case of goodness-of-fit testing in arbitrary dimension

Multivariate search for differentially expressed gene combinations

Energy distance

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