## Bernoulli

- Bernoulli
- Volume 23, Number 2 (2017), 951-989.

### Two-sample smooth tests for the equality of distributions

Wen-Xin Zhou, Chao Zheng, and Zhen Zhang

#### Abstract

This paper considers the problem of testing the equality of two unspecified distributions. The classical omnibus tests such as the Kolmogorov–Smirnov and Cramér–von Mises are known to suffer from low power against essentially all but location-scale alternatives. We propose a new two-sample test that modifies the Neyman’s smooth test and extend it to the multivariate case based on the idea of projection pursue. The asymptotic null property of the test and its power against local alternatives are studied. The multiplier bootstrap method is employed to compute the critical value of the multivariate test. We establish validity of the bootstrap approximation in the case where the dimension is allowed to grow with the sample size. Numerical studies show that the new testing procedures perform well even for small sample sizes and are powerful in detecting local features or high-frequency components.

#### Article information

**Source**

Bernoulli, Volume 23, Number 2 (2017), 951-989.

**Dates**

Received: January 2015

Revised: September 2015

First available in Project Euclid: 4 February 2017

**Permanent link to this document**

https://projecteuclid.org/euclid.bj/1486177389

**Digital Object Identifier**

doi:10.3150/15-BEJ766

**Mathematical Reviews number (MathSciNet)**

MR3606756

**Zentralblatt MATH identifier**

1380.62202

**Keywords**

goodness-of-fit high-frequency alternations multiplier bootstrap Neyman’s smooth test two-sample problem

#### Citation

Zhou, Wen-Xin; Zheng, Chao; Zhang, Zhen. Two-sample smooth tests for the equality of distributions. Bernoulli 23 (2017), no. 2, 951--989. doi:10.3150/15-BEJ766. https://projecteuclid.org/euclid.bj/1486177389