On the asymptotic normality of theL2-Distance Class of Statistics with Estimated Parameters

In the problem of testing goodness-of-fit, widely used test statistics form the L 2-distance. This paper studies the L 2-distance class of statistics for testing goodness-of-fit. The phrase “ L 2-distance class” means they consistently estimate the L 2-distance which measures the discrepancy between two probability distributions. Especially the case in which statistics include parameter estimators is investigated. It is shown that the proposed statistic has asymptotic normality under both the null and the alternative distribution. This work is essentially a generalization of the result due to Ahmad (1993) for the particular case of Cramér-von Mises statistic and is closely related to that by de Wet and Randles (1987). Several examples that illustrate the theory are also given.