The normal, Edgeworth, saddlepoint and uniform approximations to the Wilcoxon–Mann–Whitney null-distribution: a numerical comparison

Bean, Raphaël; Froda, Sorana; Eeden, Constance van

doi:10.1080/10485250310001622677

Cited by 15 publications

(9 citation statements)

References 13 publications

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“…Statisticians also focused on numerical evaluations of the asymptotic properties of the signed-rank (Wilcoxon 1947;Fellingham and Stocker 1964;McCornack 1965;Claypool and Holbert 1974) and ranksum statistics (Wilcoxon 1947;Buckle et al 1969;Di Bucchianico 1999;Bean et al 2004).…”

Section: Numerical Approachmentioning

confidence: 99%

Normal Approximations to the Distributions of the Wilcoxon Statistics: Accurate to What N ? Graphical Insights

Bellera

Julien

Hanley

2010

Journal of Statistics Education

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The Wilcoxon statistics are usually taught as nonparametric alternatives for the 1-and 2-sample Student-t statistics in situations where the data appear to arise from non-normal distributions, or where sample sizes are so small that we cannot check whether they do. In the past, critical values, based on exact tail areas, were presented in tables, often laid out in a way that saves space but makes them confusing to look up. Recently, a number of textbooks have bypassed the tables altogether, and suggested using normal approximations to these distributions, but these texts are inconsistent as to the sample size n at which the standard normal distribution becomes more accurate as an approximation. In the context of non-normal data, students can find the use of this approximation confusing. This is unfortunate given that the reasoning behind-and even the derivation of-the exact distributions 1 Journal of Statistics Education, Volume 18, Number 2, (2010) can be so easy to teach but also help students understand the logic behind rank tests. This note describes a heuristic approach to the Wilcoxon statistics. Going back to first principles, we represent graphically their exact distributions. To our knowledge (and surprise) these pictorial representations have not been shown earlier. These plots illustrate very well the approximate normality of the statistics with increasing sample sizes, and importantly, their remarkably fast convergence.

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Section: Numerical Approachmentioning

confidence: 99%

Normal Approximations to the Distributions of the Wilcoxon Statistics: Accurate to What N ? Graphical Insights

Bellera

Julien

Hanley

2010

Journal of Statistics Education

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“…Further, in Bean et al (2004) it is shown that the previous normal approximation can be improved by an Edgeworth approximation or a saddlepoint one. Unlike W 0 , the distribution of W XY under the alternative hypothesis, denoted by W FG , depends not only on the sample sizes m and n, but also on F and G. For this reason, it is more difficult to study W FG than W 0 .…”

Section: Recalling Wrs Testmentioning

confidence: 99%

On stochastic orderings of the Wilcoxon Rank Sum test statistic—With applications to reproducibility probability estimation testing

Capitani¹,

Martini²

2011

Statistics & Probability Letters

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Recently, the possibility of testing statistical hypotheses through the estimate of the reproducibility probability (i.e. the estimate of the power of the statistical test) in a general parametric framework has been introduced. In this paper, we provide some results on the stochastic orderings of the Wilcoxon Rank Sum (WRS) statistic, implying, for example, that the related test is strictly unbiased. Moreover, under some regularity conditions, we show that it is possible to define a continuous and strictly monotone power function of the WRS test. This last result is useful in order to obtain a point estimator and lower bounds for the power of the WRS test. In analogy with the parametric setting, we show that these power estimators, alias reproducibility probability estimators, can be used as test statistic, i.e. it is possible to refer directly to the estimate of the reproducibility probability to perform the WRS test. Some reproducibility probability estimators based on asymptotic approximations of the power are provided. A brief simulation shows a very high agreement between the approximated reproducibility probability based tests and the classical one.

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“…Froda and van Eeden (2000) proposed a uniform saddlepoint expansion to the null distribution of the Wilcoxon-Mann-Whitney test. Additionally, Bean, Froda and van Eeden (2004) compared a saddlepoint approximation of the Wilcoxon-MannWhitney test with that of Edgeworth, and determined normal and uniform approximations under finite sample sizes. Recently, Murakami and Kamakura (2009) considered a saddlepoint approximation to the distribution of the Jonckheere-Terpstra test (Gibbons and Chakraborti, 2003) with a continuity correction.…”

Section: Introductionmentioning

confidence: 99%

Approximations to the distribution of a combination of the Wilcoxon and Mood statistics: A numerical comparison

Murakami

2011

Journal of the Japanese Society of Computational Statistics

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On testing hypotheses in two-sample problems, the Lepage-type statistic is often used for testing the location and scale parameters. Various Lepage-type statistics have been proposed and discussed by many authors over the course of many years. One of the most famous and powerful Lepage-type statistics is a combination of the Wilcoxon and Mood statistics, namely T . Deriving the exact critical value of the T statistic is difficult when the sample sizes are increased. In that situation, an approximation to the distribution function of a test statistic is extremely important in statistics. The gamma approximation and the saddlepoint approximations are used to evaluate in upper tail probability for the T statistic under finite sample sizes. The accuracy of various approximations to the exact probability of the T statistic is investigated.

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The normal, Edgeworth, saddlepoint and uniform approximations to the Wilcoxon–Mann–Whitney null-distribution: a numerical comparison

Cited by 15 publications

References 13 publications

Normal Approximations to the Distributions of the Wilcoxon Statistics: Accurate to What N ? Graphical Insights

Normal Approximations to the Distributions of the Wilcoxon Statistics: Accurate to What N ? Graphical Insights

On stochastic orderings of the Wilcoxon Rank Sum test statistic—With applications to reproducibility probability estimation testing

Approximations to the distribution of a combination of the Wilcoxon and Mood statistics: A numerical comparison

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