## Annals of Mathematical Statistics

- Ann. Math. Statist.
- Volume 18, Number 4 (1947), 540-548.

### Some Basic Theorems for Developing Tests of Fit for The Case of the Non-Parametric Probability Distribution Function, I

#### Abstract

In developing tests of fit based upon a sample $O_n(x_i)$ in the case that the cumulative distribution function $F(X)$ of the universe of $X$'s is not necessarily a function of a finite number of specific parameters--sometimes known as the non-parametric case--it has been pointed out by several writers that the "probability integral transformation" is a useful device (cf. [1]-[4]). The author finds that a modification of this approach is more effective. This modification is to use a transformation of ordered sample values $x_i$ from a random sample $O_n(x_i)$ based on successive differences of the cdf values $F(x_i)$. A theorem is proved giving a simple formula for the expected values of the products of powers of these differences, where all differences from 1 to $n + 1$ are involved in a symmetrical manner. The moment generating function of the test function defined as the sum of $m$ squares of these successive differences is developed and the application of such a test function is briefly discussed.

#### Article information

**Source**

Ann. Math. Statist., Volume 18, Number 4 (1947), 540-548.

**Dates**

First available in Project Euclid: 28 April 2007

**Permanent link to this document**

https://projecteuclid.org/euclid.aoms/1177730344

**Digital Object Identifier**

doi:10.1214/aoms/1177730344

**Mathematical Reviews number (MathSciNet)**

MR23037

**Zentralblatt MATH identifier**

0029.15402

**JSTOR**

links.jstor.org

#### Citation

Kimball, Bradford F. Some Basic Theorems for Developing Tests of Fit for The Case of the Non-Parametric Probability Distribution Function, I. Ann. Math. Statist. 18 (1947), no. 4, 540--548. doi:10.1214/aoms/1177730344. https://projecteuclid.org/euclid.aoms/1177730344