A Nonparametric Test of Independence between Two Vectors

Abstract

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.