Tests Based on Linear Combinations of the Orthogonal Components of the Cramer-von Mises Statistic When Parameters are Estimated

Abstract

In a previous work, the author showed how linear combinations of the orthogonal components of the Cramer-von Mises statistic could be used to test fit to a fully specified distribution function. In this paper, the results are extended to the case where $r$ parameters are estimated from the data. It is shown that if the coefficient vector of the linear combination is orthogonal to a specified $r$ dimensional subspace, then the asymptotic distribution of that combination is the same whether the parameters are estimated or known exactly.