## The Annals of Mathematical Statistics

### On the Power of Certain Tests for Independence in Bivariate Populations

H. S. Konijn

#### Abstract

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

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

Dates
First available in Project Euclid: 28 April 2007

https://projecteuclid.org/euclid.aoms/1177728260

Digital Object Identifier
doi:10.1214/aoms/1177728260

Mathematical Reviews number (MathSciNet)
MR79384

Zentralblatt MATH identifier
0075.29302

JSTOR