Components of Cramér–Von Mises Statistics. I

TLDR

The sample distribution function Fn(x) is derived from independent uniform (0, 1) variables. The study aims to express the Cramér‑von Mises statistic Wn² as an orthogonal sum of principal components z_nj² weighted by (jπ)⁻². This is achieved by decomposing n{Fn(x)−x} into its orthogonal principal components and also exploring an alternative component set from Pyke’s modified distribution function. The resulting components z_nj are identically distributed, their significance points are tabulated, and tests based on them show asymptotic powers comparable to Wn², An², and Un²; residual‑statistic significance points are provided, the components relate to Legendre polynomials in Neyman’s smooth test, and the method is applied to coal‑mining disaster data.

Abstract

Summary Let Fn(x) be the sample distribution function derived from a sample of independent uniform (0, 1) variables. The paper is mainly concerned with the orthogonal representation of the Cramér-von Mises statistic Wn2 in the form ∑j=1∞(jπ)−2znj2 where the znj are the principal components of n{Fn(x)−x}. It is shown that the znj are identically distributed for each n and their significance points are tabulated. Their use for testing goodness of fit is discussed and their asymptotic powers are compared with those of Wn2, Anderson and Darling's statistic An2 and Watson's Un2 against shifts of mean and variance in a normal distribution. The asymptotic significance points of the residual statistic Wn2−∑j=1p(jπ)−2znj2 are also given for various p. It is shown that the components analogous to znj for An2 are the Legendre polynomial components introduced by Neyman as the basis for his “smooth” test of goodness of fit. The relationship of the components to a Fourier series analysis of Fn(x) –x is discussed. An alternative set of components derived from Pyke's modification of the sample distribution function is considered. Tests based on the components znj are applied to data on coalmining disasters.