A note on weighted least square distribution fitting and full standardization of the empirical distribution function

The relationship between the norm square of the standardized cumulative distribution and the chi-square statistic is examined using the form of the covariance matrix as well as the projection perspective. This investigation enables us to give uncorrelated components of the chi-square statistic and to provide interpretation of these components as innovations standardizing the cumulative distribution values. The norm square of the standardized difference between empirical and theoretical cumulative distributions is also examined as an objective function for parameter estimation. Its relationship to the chi-square distance enables us to discuss the large sample properties of these estimators and a difference in their properties in the cases that the distribution is evaluated at fixed and random points.