The X2-goodness-of-fit tests with moment type estimators
In the x2-goodness-of-fit test the underlying null hypothesis usually involves unknown parameters. In this article we study the asymptotic distribution of the Pearson statistic when the unknown parameters are estimated by a moment type estimator based on the ungrouped data. As is expected the usual Pearson statistic is no longer asymptotically x2-distributed in this situation. We propose a statistic [Qcirc] which under certain regularity conditions is asymptotically x2-distributed. We also propose a statistic Q∗ for the goodness-of-fit test when the class boundaries are random. The asymptotic powers of [Qcirc] and [Qcirc]∗ tests are discussed.