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
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. -). 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.
Ann. Math. Statist., Volume 18, Number 4 (1947), 540-548.
First available in Project Euclid: 28 April 2007
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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