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December, 1947 Some Basic Theorems for Developing Tests of Fit for The Case of the Non-Parametric Probability Distribution Function, I
Bradford F. Kimball
Ann. Math. Statist. 18(4): 540-548 (December, 1947). DOI: 10.1214/aoms/1177730344

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.

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Bradford F. Kimball. "Some Basic Theorems for Developing Tests of Fit for The Case of the Non-Parametric Probability Distribution Function, I." Ann. Math. Statist. 18 (4) 540 - 548, December, 1947. https://doi.org/10.1214/aoms/1177730344

Information

Published: December, 1947
First available in Project Euclid: 28 April 2007

zbMATH: 0029.15402
MathSciNet: MR23037
Digital Object Identifier: 10.1214/aoms/1177730344

Rights: Copyright © 1947 Institute of Mathematical Statistics

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Vol.18 • No. 4 • December, 1947
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