Standardizing the Empirical Distribution Function Yields the Chi-Square Statistic

Standardizing the empirical distribution function yields a statistic with norm square that matches the chi-square test statistic. To show this one may use the covariance matrix of the empirical distribution which, at any finite set of points, is shown to have an inverse which is tridiagonal. Moreover, a representation of the inverse is given which is a product of bidiagonal matrices corresponding to a representation of the standardization of the empirical distribution via a linear combination of values at two consecutive points. These properties are discussed also in the context of minimum distance estimation.