A Note on Testing Separability in Spatial‐Temporal Marked Point Processes

Abstract

In environmental risk analysis, it is common to assume the stochastic independence (or separability) between the marks associated with the random events of a spatial-temporal point process. Schoenberg (2004, Biometrics 60, 471-481) proposed several test statistics for this hypothesis and used simulated data to evaluate their performance. He found that a Cramér-von Mises-type test is powerful to detect gradual departures from separability although it is not uniformly powerful over a large class of alternative models. We present a semiparametric approach to model alternative hypotheses to separability and derive a score test statistic. We show that there is a relationship between this score test and some of the test statistics proposed by Schoenberg. Specifically, all are different versions of weighted Cramér-von Mises-type statistics. This gives some insight into the reasons for the similarities and differences between the test statistics' performance. We also point out some difficulties in controlling the type I error probability in Schoenberg's residual test.