The Annals of Mathematical Statistics

On the Power of Certain Tests for Independence in Bivariate Populations

H. S. Konijn

Full-text: Open access


Let $F_{\lambda^0}$ denote the joint distribution of two independent random variables $Y_{\lambda^0}$ and $Z_{\lambda^0}$. The paper investigates properties of the joint distribution $F_\lambda$ of the linearly transformed random variables $Y_\lambda$ and $Z_\lambda$. Let $\Im_0$ be the Spearman rank correlation test, $\Im_1$ the difference sign correlation test, $\Im_2$ the unbiased grade correlation test (which is asymptotically equivalent to $\Im_0$), $\Im_3$ the medial correlation test, and $\mathcal{R}$ the ordinary (parametric) correlation test. (Whenever discussing $\mathcal{R}$ we assume existence of fourth moments.) Properties of the power of these tests are found for alternatives of the above-mentioned form, particularly for alternatives "close" to the hypothesis of independence and for large samples. Against these alternatives the efficiency of $\Im_3$ is found to depend strongly on local properties of the densities of $Y_{\lambda_0}$ and $Z_{\lambda^0}$, which should invite caution; and the efficiency of $\Im_1$ with respect to $\Im_0$ is often unity. Incidentally, Pitman's result on efficiency is extended in several directions.

Article information

Ann. Math. Statist., Volume 27, Number 2 (1956), 300-323.

First available in Project Euclid: 28 April 2007

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier



Konijn, H. S. On the Power of Certain Tests for Independence in Bivariate Populations. Ann. Math. Statist. 27 (1956), no. 2, 300--323. doi:10.1214/aoms/1177728260.

Export citation


  • See Correction: H. S. Konijn. Correction to "On the Power of Certain Tests for Independence in Bivariate Populations". Ann. Math. Statist., Vol. 29, Iss. 3 (1958), 935--936.