Pearson chi-squared test of fit with random intervals

A modified form of the Pearson x2 test statistic is considered where estimators based on the ungrouped sample are employed in the test statistic as well as in determining the class interval end points. In the case of continuous distributions with location and scale parameters, it is shown here that the null distribution of such a modified test statistic does not depend on the parameters if these are estimated by sample mean and variance. By utilizing known results concerning the distribution of a weighted sum of independent chi-squared variates, a table of certain percentage points of the asymptotic distribution of this modified test statistic is developed in order to facilitate its use for testing normality.