Components of Cramér-Von Mises Statistics. Ii

Abstract

Summary Let θ^ be an asymptotically efficient estimator of parameter vector θ based on an ordered sample of n independent observations y 1, ..., y n from a distribution with distribution function F(y, θ) and let x^j=F(yj,θ^), j = 1, ..., n. Let Ŵn 2 be the Cramér–von Mises statistic calculated from ◯ 1, ..., ◯ n. This paper shows how to construct components ẑ n1, ẑ n2, ... such that Ŵ n 2 = ∑j=1 ∞ μj ẑ nj 2 where ẑ n1, ẑ n2, ... are asymptotically independent N(0, 1) variables and considers the use of these components for testing goodness of fit in the presence of unknown parameters. Asymptotic significance points are given for tests of normality and exponentiality for Ŵn 2, Anderson and Darling's  n 2 and Watson's Û n 2. The asymptotic powers of these statistics and of the components are studied against alternatives of interest.