## Electronic Journal of Statistics

- Electron. J. Statist.
- Volume 5 (2011), 342-372.

### Asymptotic independence of correlation coefficients with application to testing hypothesis of independence

Zhengjun Zhang, Yongcheng Qi, and Xiwen Ma

#### Abstract

This paper first proves that the sample based Pearson’s product-moment correlation coefficient and the quotient correlation coefficient are asymptotically independent, which is a very important property as it shows that these two correlation coefficients measure completely different dependencies between two random variables, and they can be very useful if they are simultaneously applied to data analysis. Motivated from this fact, the paper introduces a new way of combining these two sample based correlation coefficients into maximal strength measures of variable association. Second, the paper introduces a new marginal distribution transformation method which is based on a rank-preserving scale regeneration procedure, and is distribution free. In testing hypothesis of independence between two continuous random variables, the limiting distributions of the combined measures are shown to follow a max-linear of two independent *χ*^{2} random variables. The new measures as test statistics are compared with several existing tests. Theoretical results and simulation examples show that the new tests are clearly superior. In real data analysis, the paper proposes to incorporate nonlinear data transformation into the rank-preserving scale regeneration procedure, and a conditional expectation test procedure whose test statistic is shown to have a non-standard limit distribution. Data analysis results suggest that this new testing procedure can detect inherent dependencies in the data and could lead to a more meaningful decision making.

#### Article information

**Source**

Electron. J. Statist., Volume 5 (2011), 342-372.

**Dates**

First available in Project Euclid: 10 May 2011

**Permanent link to this document**

https://projecteuclid.org/euclid.ejs/1305034906

**Digital Object Identifier**

doi:10.1214/11-EJS610

**Mathematical Reviews number (MathSciNet)**

MR2802047

**Zentralblatt MATH identifier**

1274.62401

**Subjects**

Primary: 62H20: Measures of association (correlation, canonical correlation, etc.) 62G10: Hypothesis testing

Secondary: 62G30: Order statistics; empirical distribution functions

**Keywords**

Dependence measures rank-preserving scale regeneration distribution free test Fisher’s Z-transformation tests gamma tests χ2 tests order statistics conditional expectation

#### Citation

Zhang, Zhengjun; Qi, Yongcheng; Ma, Xiwen. Asymptotic independence of correlation coefficients with application to testing hypothesis of independence. Electron. J. Statist. 5 (2011), 342--372. doi:10.1214/11-EJS610. https://projecteuclid.org/euclid.ejs/1305034906