“Large Sample-Size Distribution of Statistics Used in Testing for Spatial Correlation”: A Reply

TLDR: In this article, the authors proved a general theorem on the large sample normality of quadratic forms and showed that the conditions given in a similar theorem (Cliff and Ord) are inadequate to ensure normality.

Abstract: Test statistics for testing for spatial correlation in continuous variables have been given by both Moran and Geary and have subsequently been generalized. It has been conjectured for a long time that under the hypothesis of no spatial correlations all these statistics are normally distributed when the sample size is large. This paper proves a very general theorem on the large sample normality of quadratic forms. As corollaries to the theorem the asymptotic normality, under the hypothesis, of all the above-mentioned statistics is established. The necessary conditions are quite unrestrictive. It is also shown, by means of a counter example, that the conditions given in a similar theorem (Cliff and Ord) are inadequate to ensure normality.