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

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. 1. Introduction. The orthogonal components of the Cramer-von Mises statistic were introduced by Durbin and Knott (1972) to test goodness of fit to a completely specified distribution function. Schoenfeld (1977) examined the asymptotic properties of linear combinations of a generalization of these components. He defined a contiguous family of alternative distributions and showed that for any member of this family an asymptotically most powerful test could be found based on a linear combination of the components. The asymptotic power and efficiency of these tests were shown to have simple expressions. This paper extends Schoenfeld's results to the case where r parameters of the hypothetical distribution function are estimated from the data. The components can be computed using estimates of the parameters. If the coefficient vector of a linear combination of the components is orthogonal to a specified r dimensional subspace, then the asymptotic distribution of that combination is the same as if the true parameter values were used. This theory is applied to the case where location and scale parameters are unknown.