Nonparametric Tests for Scale

Klotz, Jerome

doi:10.1214/aoms/1177704576

Cited by 124 publications

(30 citation statements)

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“…Since the asymptotic normality of the test statistic of the Ansari-Bradley test and the Mood test under the alternative hypothesis have been examined intensely (cf., e.g., [103,104] [104][105][106]. The asymptotic relative efficiency (ARE) with respect to the traditional F-test for differences in scale for two Gaussian distributions has been discussed by [103].…”

Section: Two-sample Rank Test On Dispersionmentioning

confidence: 99%

Cumulative Paired φ-Entropy

Klein

Mangold

Doll

2016

Entropy

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Abstract:A new kind of entropy will be introduced which generalizes both the differential entropy and the cumulative (residual) entropy. The generalization is twofold. First, we simultaneously define the entropy for cumulative distribution functions (cdfs) and survivor functions (sfs), instead of defining it separately for densities, cdfs, or sfs. Secondly, we consider a general "entropy generating function" φ, the same way Burbea et al. (IEEE Trans. Inf. Theory 1982, 28, 489-495) and Liese et al. (Convex Statistical Distances; Teubner-Verlag, 1987) did in the context of φ-divergences. Combining the ideas of φ-entropy and cumulative entropy leads to the new "cumulative paired φ-entropy" (CPE φ ). This new entropy has already been discussed in at least four scientific disciplines, be it with certain modifications or simplifications. In the fuzzy set theory, for example, cumulative paired φ-entropies were defined for membership functions, whereas in uncertainty and reliability theories some variations of CPE φ were recently considered as measures of information. With a single exception, the discussions in the scientific disciplines appear to be held independently of each other. We consider CPE φ for continuous cdfs and show that CPE φ is rather a measure of dispersion than a measure of information. In the first place, this will be demonstrated by deriving an upper bound which is determined by the standard deviation and by solving the maximum entropy problem under the restriction of a fixed variance. Next, this paper specifically shows that CPE φ satisfies the axioms of a dispersion measure. The corresponding dispersion functional can easily be estimated by an L-estimator, containing all its known asymptotic properties. CPE φ is the basis for several related concepts like mutual φ-information, φ-correlation, and φ-regression, which generalize Gini correlation and Gini regression. In addition, linear rank tests for scale that are based on the new entropy have been developed. We show that almost all known linear rank tests are special cases, and we introduce certain new tests. Moreover, formulas for different distributions and entropy calculations are presented for CPE φ if the cdf is available in a closed form.

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Section: Two-sample Rank Test On Dispersionmentioning

confidence: 99%

Cumulative Paired φ-Entropy

Klein

Mangold

Doll

2016

Entropy

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“…Since the numerical values for http://www.jsdajournal.com/content/1/1/6 assessing the probabilities in Equation (1) are considerably complicated in computation when F and G are continuous distributions with F = G. As the rank-sum test is widely adopted for testing the center differences of two distributions, it is natural to study the efficiency of a rank-sum test for variability (Ansari and Bradley 1960). For decades, studies have focused on proposing new definitions of the rank statistic and using the methods of Chernoff and Savage to show the relative efficiency of the proposed statistic to the F-test, see for example Mood (1954), Siegel and Tukey (1960), Ansari and Bradley (1960), and Klotz (1962). Ansari and Bradley (1960) mentioned that if the means of the X and Y samples cannot be considered equal, differences in location have a severe impact on all the tests of dispersion.…”

Section: E(u Xmentioning

confidence: 99%

“…Ansari and Bradley (1960) mentioned that if the means of the X and Y samples cannot be considered equal, differences in location have a severe impact on all the tests of dispersion. Klotz (1962) showed the power of a rank test can be found by integrating the joint density of X and Y samples over that part of the m + n dimensional space defined by the alternative orderings which lie in the critical region of the test, for which conditions are very strict.…”

Section: E(u Xmentioning

confidence: 99%

Joint distribution of rank statistics considering the location and scale parameters and its power study

Lee

2014

J Stat Distrib Appl

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The ranking method used for testing the equivalence of two distributions has been studied for decades and is widely adopted for its simplicity. However, due to the complexity of calculations, the power of the test is either estimated by a normal approximation or found when an appropriate alternative is given. Here, via the Finite Markov chain imbedding technique, we are able to establish the marginal and joint distributions of the rank statistics considering the shift and scale parameters, respectively and simultaneously, under two different continuous distribution functions. Furthermore, the procedures of distribution equivalence tests and their power functions are discussed. Numerical results of a joint distribution of rank statistics under the standard normal distribution and the powers for a sequence of alternative normal distributions with means from −20 to 20 and standard deviations from 1 to 9 and their reciprocal are presented. In addition, we discuss the powers of the rank statistics under the Lehmann alternatives.

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“…These tests have however been shown to be equivalent [10]. Also, all such tests require some assumption on the kind of admissible distribution pairs to be compared, for instance that they should have the same median [18].…”

Section: Rank Statistics and Pixelsmentioning

confidence: 99%

Variance Ranklets: orientation–selective rank features for contrast modulations

Azzopardi¹,

Smeraldi²

2009

Procedings of the British Machine Vision Conference 2009

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We introduce a novel type of orientation-selective rank features that are sensitive to contrast modulations (second-order stimuli). Variance Ranklets are designed in close analogy with the standard Ranklets, but use the Siegel-Tukey statistics for dispersion instead of the Wilcoxon statistics. Their response shows the same orientation selectivity pattern of Haar wavelets on second-order signals that are not detectable by linear filters. To the best of our knowledge, this is the first family of rank filters designed to detect orientation in variance modulations.We validate our descriptors with an application to texture classification over a subset of the VisTex and Brodatz databases. The combination of standard (intensity) Ranklets with Variance Ranklets greatly improves on the performance of Ranklets alone. Comparison with other published results shows that state-of-the-art recognition rates can be achieved with a simple Nearest Neighbour classifier.

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Nonparametric Tests for Scale

Cited by 124 publications

References 0 publications

Cumulative Paired φ-Entropy

Cumulative Paired φ-Entropy

Joint distribution of rank statistics considering the location and scale parameters and its power study

Variance Ranklets: orientation–selective rank features for contrast modulations

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