# UCL Discovery

## Components of Cramer-von Mises statistics. I

Durbin, J.; Knott, M.; (1972) Components of Cramer-von Mises statistics. I. Journal of the Royal Statistical Society: Series B (Methodological) , 34 (2) pp. 290-307.

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## Abstract

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

Type: Article Components of Cramer-von Mises statistics. I http://www.rss.org.uk/main.asp?page=1711 English UCL > School of Arts and Social Sciences > Faculty of Social and Historical Sciences > Economics http://discovery.ucl.ac.uk/id/eprint/18452

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