Large Sample-Size Distribution of Statistics Used In Testing for Spatial Correlation

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.