Least-Squares Approximation to Minimum Chi-Square Estimators of Location and Scale Parameters and Their Effect on the Pearson Chi-Square Test.

Abstract : Application of the Pearson chi-square test to goodness of fit of a distribution often leads to serious difficulties, particularly in the formation of intervals (as in the case of a continuous distribution) and in the estimation of unknown parameters. Under suitable conditions and with appropriately constructed estimators of the parameters, the test statistic converges in distribution to that of chi-square as the sample size increases. In the present paper, a comparatively simple least-squares approximation to the minimum chi-square estimator is developed which, when appropriately implemented, results in an asymptotic chi-square distribution of the test statistic. This estimator is developed for the cases of fixed and random intervals, and the role of the underlying assumptions is studied in detail. keywords: Pearson chi-square test, test of fit, asymptotic distribution, least-squares approximation.