## The Annals of Mathematical Statistics

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

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

#### 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

**Permanent link to this document**

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**

links.jstor.org

#### Citation

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

#### Corrections

- 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.Project Euclid: euclid.aoms/1177706558