Nonparametric test of independence between two vectors

A new statistic, (Q) over cap(n)$, based on interdirections is proposed for testing whether two vector-valued quantities are dependent. The statistic, which has an intuitive invariance property, reduces to the quadrant statistic when the two quantities are each univariate. Under the null hypothesis of independence, (Q) over cap(n)$, 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 (Q) over cap(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 (Q) over cap(n)$ performs better than the others for heavy-tailed distributions and is competitive for distributions with moderate rail weights. Finally, an example illustrates that (Q) over cap(n)$ is resistant to outliers.