## The Annals of Statistics

- Ann. Statist.
- Volume 1, Number 1 (1973), 54-66.

### On the Centering of a Simple Linear Rank Statistic

#### Abstract

Hajek (1968) proved that under weak conditions the distribution of a simple linear rank statistic $S$ is asymptotically normal, centered at the mean $ES$. He left open the question whether under the same conditions the centering constant $ES$ may be replaced by a simpler constant $\mu$, as was found to be true in the two-sample case and under different conditions by Chernoff and Savage (1958) and Govindarajulu, LeCam and Rhagavachari (1966). In this paper it is shown that the replacement of $ES$ by $\mu$ is permissible if one of Hajek's conditions is slightly strengthened.

#### Article information

**Source**

Ann. Statist., Volume 1, Number 1 (1973), 54-66.

**Dates**

First available in Project Euclid: 25 October 2007

**Permanent link to this document**

https://projecteuclid.org/euclid.aos/1193342381

**Digital Object Identifier**

doi:10.1214/aos/1193342381

**Mathematical Reviews number (MathSciNet)**

MR362689

**Zentralblatt MATH identifier**

0255.62015

**Subjects**

Primary: 62G10: Hypothesis testing

Secondary: 60F05: Central limit and other weak theorems

**Keywords**

Simple linear rank statistic asymptotically normal centering constant

#### Citation

Hoeffding, Wassily. On the Centering of a Simple Linear Rank Statistic. Ann. Statist. 1 (1973), no. 1, 54--66. doi:10.1214/aos/1193342381. https://projecteuclid.org/euclid.aos/1193342381