## The Annals of Statistics

### Graph-Theoretic Measures of Multivariate Association and Prediction

#### Abstract

Interpoint-distance-based graphs can be used to define measures of association that extend Kendall's notion of a generalized correlation coefficient. We present particular statistics that provide distribution-free tests of independence sensitive to alternatives involving non-monotonic relationships. Moreover, since ordering plays no essential role, the ideas are fully applicable in a multivariate setting. We also define an asymmetric coefficient measuring the extent to which (a vector) $X$ can be used to make single-valued predictions of (a vector) $Y$. We discuss various techniques for proving that such statistics are asymptotically normal. As an example of the effectiveness of our approach, we present an application to the examination of residuals from multiple regression.

#### Article information

Source
Ann. Statist. Volume 11, Number 2 (1983), 377-391.

Dates
First available in Project Euclid: 12 April 2007

http://projecteuclid.org/euclid.aos/1176346148

JSTOR

Digital Object Identifier
doi:10.1214/aos/1176346148

Mathematical Reviews number (MathSciNet)
MR696054

Zentralblatt MATH identifier
0528.62052

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

#### Citation

Friedman, Jerome H.; Rafsky, Lawrence C. Graph-Theoretic Measures of Multivariate Association and Prediction. Ann. Statist. 11 (1983), no. 2, 377--391. doi:10.1214/aos/1176346148. http://projecteuclid.org/euclid.aos/1176346148.