On a Cramér-von Mises-Type Statistic for Testing Symmetry
James A. Koziol · Journal of the American Statistical Association · 1980
A Cramér-von Mises-type statistic for testing symmetry, proposed independently by Orlov (1972) and Rothman and Woodroofe (1972), is decomposed into components in the manner of Durbin and Knott (1972); the components are found to be related to Fourier series expansions of the underlying distribution function. Asymptotic power properties of the statistic and its components in testing symmetry about the origin against location shift alternatives when observations have double exponential or normal distributions are described. From these power considerations, it is suggested that the first component is a more useful statistic for the testing problem than the overall Cramér-von Mises-type statistic. The components are shown to be asymptotically equivalent to linear rank statistics. On the basis of computational convenience, a new linear rank statistic for testing symmetry, the analog of the first component, is proposed.