Testing spatial randomness based on empirical distribution function: A study on lattice data

In this article, we extended the empirical distribution function based test statistic _Ik$I_k$ of Skaug and Tjostheim [1993. Nonparametric test of serial independence based on the empirical distribution function. Biometrika 80, 591–602] in the time series setting to _Dn$D_n$ for spatial lattice data and derived the asymptotic distribution of the proposed test statistic _Dn$D_n$ under the null hypothesis of spatial independence. The size and power of the proposed test statistic under conditional autoregressive model (CAR) were simulated. We applied _Dn$D_n$, Moran's I and Geary's c   to the transformed and well-studied sudden infant death syndrome data from North Carolina and found that _Dn$D_n$ produced a much smaller p$p$-value in testing spatial independence.