## The Annals of Statistics

- Ann. Statist.
- Volume 4, Number 3 (1976), 554-575.

### Weak Convergence of Sequential Linear Rank Statistics

#### Abstract

A sequential version of Chernoff-Savage linear rank statistics is introduced as a basis for inference. The principal result is an invariance principle for two-sample rank statistics, i.e., under a fixed alternative the sequence of sequential linear rank statistics converges weakly to a Wiener process. The domain of application of the theorem is quite broad and includes score functions which tend to infinity at the end points much more rapidly than that of the normal scores test. The method of proof involves new results in the theory of multiparameter empirical processes as well as some new probability bounds on the joint behavior of uniform order statistics. Applications of weak convergence are explored; in particular, the extension of the theory of Pitman efficiency to the sequential case.

#### Article information

**Source**

Ann. Statist., Volume 4, Number 3 (1976), 554-575.

**Dates**

First available in Project Euclid: 12 April 2007

**Permanent link to this document**

https://projecteuclid.org/euclid.aos/1176343461

**Digital Object Identifier**

doi:10.1214/aos/1176343461

**Mathematical Reviews number (MathSciNet)**

MR403017

**Zentralblatt MATH identifier**

0331.62017

**JSTOR**

links.jstor.org

**Subjects**

Primary: 62E20: Asymptotic distribution theory

Secondary: 62G10: Hypothesis testing 62G20: Asymptotic properties 62L10: Sequential analysis

**Keywords**

Sequential linear rank statistics weak convergence uniform order statistics asymptotic efficiency

#### Citation

Braun, Henry I. Weak Convergence of Sequential Linear Rank Statistics. Ann. Statist. 4 (1976), no. 3, 554--575. doi:10.1214/aos/1176343461. https://projecteuclid.org/euclid.aos/1176343461