A Nonparametric Test of Independence between Two Vectors

• Published in: Journal of the American Statistical Association Vol. 92; no. 438; pp. 561 - 567
• Main Authors: Gieser, Peter W., Randles, Ronald H.
• Format: Journal Article
• Language: English
• Published: Alexandria, VA Taylor & Francis Group 01.06.1997; American Statistical Association; Taylor & Francis Ltd
• Subjects: Approximation; Association; Componentwise operations; Correlation coefficients; Distribution theory; Exact sciences and technology; Independence test; Interdirection; Mathematics; Matrices; Monte Carlo; Nonparametric inference; Nonparametric tests; Null hypothesis; Pitman efficiency; Probability and statistics; Quadrants; Random variables; Robust; Sciences and techniques of general use; Statistical median; Statistical methods; Statistics; Tests; Theory and Methods; Vector independence; Monte Carlo method; Non parametric test; Independence; Likelihood ratio; Quadrant statistic; Asymptotic efficiency; Relative efficiency; Chi square; Elliptic symmetry; Test; Independence test; Numerical simulation; Invariance property; Vector; Wilks criterion
• ISSN: 0162-1459; 1537-274X
• DOI: 10.1080/01621459.1997.10474008

A new statistic, [Qcirc] n , based on interdirections is proposed for testing whether two vector-valued quantities are dependent. The statistic, which has an intuitive invariance property, reduces to the quadrant statistic when the two quantities are each univariate. Under the null hypothesis of independence, [Qcirc] n has a limiting chi-squared distribution when each vector is elliptically symmetric. The new statistic is compared to the classical normal theory competitor-Wilks' likelihood ratio criterion-and a componentwise quadrant statistic. Using a novel model of dependence between the vectors, Pitman asymptotic relative efficiencies (ARE's) are computed. The Pitman ARE's indicate that [Qcirc] n compares favorably to Wilks' likelihood ratio criterion when the vectors have heavy-tailed elliptically symmetric distributions and is uniformly better than the componentwise quadrant statistic when the vectors are spherically symmetric. A simulation study demonstrates that [Qcirc] n performs better than the others for heavy-tailed distributions and is competitive for distributions with moderate tail weights. Finally, an example illustrates that [Qcirc] n is resistant to outliers.