Revisiting the random shift approach for testing in spatial statistics

We consider the problem of non-parametric testing independence two components a stationary bivariate spatial process. In particular, we revisit random shift approach that has become standard method for independent superposition hypothesis in statistics, and it is widely used plethora practical applications. However, this liberality caused by breaking marginal correlation structure due to toroidal correction. This indeed causes assumption exchangability, which essential Monte Carlo test be exact, not fulfilled. present number permutation strategies show with variance correction brings suitable improvement compared torus field case. It reduces achieves largest power from all investigated variants. To obtain method, several approaches were studied. The best results achieved, sample covariance as factor $1/n$. corresponds asymptotic order statistics. point process case, deviations exchangeability far more complex propose an alternative strategy based on mean cross nearest-neighbor distance but slightly lower than usual $K$-function. Therefore recommend it, when patterns are clustered, where $K$-function liberality.