Components of Cramer-von Mises statistics. I

TLDR: In this article, the orthogonal representation of the Cramer-von Mises statistic W 2 n in the form Σ ∞ j=1 (jπ) -2 z 2 nj where the z nj are the principal components of $\sqrt n\{F_n(x) - x\}$.

Abstract: Let F n (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 Cramer-von Mises statistic W 2 n in the form Σ ∞ j=1 (jπ) -2 z 2 nj where the z nj are the principal components of $\sqrt n\{F_n(x) - x\}$ . It is shown that the z nj 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 W 2 n , Anderson and Darling's statistic A 2 n and Watson's U 2 n against shifts of mean and variance in a normal distribution. The asymptotic significance points of the residual statistic W 2 n - Σ p j=1 (jπ) -2 z 2 nj are also given for various p. It is shown that the components analogous to z nj for A 2 n 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 F n (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 z nj are applied to data on coal-mining disasters.