## Brazilian Journal of Probability and Statistics

- Braz. J. Probab. Stat.
- Volume 33, Number 3 (2019), 425-454.

### A rank-based Cramér–von-Mises-type test for two samples

Jamye Curry, Xin Dang, and Hailin Sang

#### Abstract

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.

#### Article information

**Source**

Braz. J. Probab. Stat., Volume 33, Number 3 (2019), 425-454.

**Dates**

Received: August 2017

Accepted: February 2018

First available in Project Euclid: 10 June 2019

**Permanent link to this document**

https://projecteuclid.org/euclid.bjps/1560153846

**Digital Object Identifier**

doi:10.1214/18-BJPS396

**Mathematical Reviews number (MathSciNet)**

MR3960270

**Zentralblatt MATH identifier**

07094811

**Keywords**

Cramér–von Mises criterion Hájek projection nonparametric test rank two-sample test

#### Citation

Curry, Jamye; Dang, Xin; Sang, Hailin. A rank-based Cramér–von-Mises-type test for two samples. Braz. J. Probab. Stat. 33 (2019), no. 3, 425--454. doi:10.1214/18-BJPS396. https://projecteuclid.org/euclid.bjps/1560153846