## The Annals of Mathematical Statistics

- Ann. Math. Statist.
- Volume 35, Number 1 (1964), 150-156.

### Some Basic Theorems of Distribution-Free Statistics

#### Abstract

The object of this article is the analysis of the concept of distribution-freeness. In Section 2 the common types of distribution-free (DF) statistics are defined and related. In Section 3 a generalization of the invariance principle is employed to construct DF statistics for classes of cpfs generated by groups of transformations of the sample space. It is shown in Section 4 that a function of several DF statistics is not necessarily DF, but that independence guarantees the desired result. The existence of a DF statistic is shown, in Section 5, to be equivalent to the existence of a suitable partition of the sample space. Several open problems are mentioned in Section 6, the final section.

#### Article information

**Source**

Ann. Math. Statist., Volume 35, Number 1 (1964), 150-156.

**Dates**

First available in Project Euclid: 27 April 2007

**Permanent link to this document**

https://projecteuclid.org/euclid.aoms/1177703736

**Digital Object Identifier**

doi:10.1214/aoms/1177703736

**Mathematical Reviews number (MathSciNet)**

MR158480

**Zentralblatt MATH identifier**

0136.40502

**JSTOR**

links.jstor.org

#### Citation

Bell, C. B. Some Basic Theorems of Distribution-Free Statistics. Ann. Math. Statist. 35 (1964), no. 1, 150--156. doi:10.1214/aoms/1177703736. https://projecteuclid.org/euclid.aoms/1177703736