The Moran coefficient for non-normal data

This paper summarizes findings that extend statistical distribution properties of the Moran coefficient index measuring spatial autocorrelation to non-normal random variables. Pitman–Koopmans theorem results are extended for the mean and the variance of this index. This summary includes a corollary to this theorem, as well as a new theorem (with its proof) and two conjectures implied by it. The first of these statements is supported by asymptotic heuristics; the second is supported by simulation experiment results. Mixture random variables that include heteroscedasticity or overdispersion also are explored. In addition, a simple asymptotic variance for the Moran coefficient is presented, assessed, and found to be very precise for sample sizes as small as 25–100. The principal conclusion is that independence and sample size are the most relevant properties for Pitman–Koopmans theorem results to be extended to non-normal random variables. The independent and identically distributed property reduces the necessary sample size for this extension, as do the properties of symmetry and normal approximation.