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.