Cramér-Von Mises Statistics for Discrete Distributions

The Cramér-von Mises family of goodness-of-fit statistics is a well-known group of statistics used to test fit to a continuous distribution. In this article we extend the family to provide tests for discrete distributions. The statistics examined are the analogues of those associated with the names of Cramér-von Mises, Watson and Anderson-Darling, called W , U and A respectively, and their components. We provide formulas for the test statistics, and asymptotic percentage points for the test for a uniform distribution with k cells. The tests are based on the empirical distribution function (EDF) of the sample. They are closely related to Pearson’s X test, and to Neyman-Barton smooth tests; in particular, all the tests can be broken down into components, as has been observed by many authors. It is suggested that A be used to test the overall null hypothesis in general, and U for the particular case where observations are counts around a circle. Their components can be used to test for particular types of departure from the null. In section 2, we define the test statistics and give the general distribution theory. In section 3 the solution of the uniform case is given, together with