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August 2019 A rank-based Cramér–von-Mises-type test for two samples
Jamye Curry, Xin Dang, Hailin Sang
Braz. J. Probab. Stat. 33(3): 425-454 (August 2019). DOI: 10.1214/18-BJPS396


We study a rank based univariate two-sample distribution-free test. The test statistic is the difference between the average of between-group rank distances and the average of within-group rank distances. This test statistic is closely related to the two-sample Cramér–von Mises criterion. They are different empirical versions of a same quantity for testing the equality of two population distributions. Although they may be different for finite samples, they share the same expected value, variance and asymptotic properties. The advantage of the new rank based test over the classical one is its ease to generalize to the multivariate case. Rather than using the empirical process approach, we provide a different easier proof, bringing in a different perspective and insight. In particular, we apply the Hájek projection and orthogonal decomposition technique in deriving the asymptotics of the proposed rank based statistic. A numerical study compares power performance of the rank formulation test with other commonly-used nonparametric tests and recommendations on those tests are provided. Lastly, we propose a multivariate extension of the test based on the spatial rank.


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Jamye Curry. Xin Dang. Hailin Sang. "A rank-based Cramér–von-Mises-type test for two samples." Braz. J. Probab. Stat. 33 (3) 425 - 454, August 2019.


Received: 1 August 2017; Accepted: 1 February 2018; Published: August 2019
First available in Project Euclid: 10 June 2019

zbMATH: 07094811
MathSciNet: MR3960270
Digital Object Identifier: 10.1214/18-BJPS396

Rights: Copyright © 2019 Brazilian Statistical Association


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Vol.33 • No. 3 • August 2019
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