Testing for a covariate effect in a parametric point process model is usually done through the Wald test, which relies on an asymptotic null distribution of the test statistic. We propose a Monte Carlo version of the test that also allows local investigation of the covariate effect in the globally fitted model. Two different test statistics are suggested for this purpose: the first, a spatial statistic computed at every location of the observation window, resembles the classical $F$-statistic that is usually used in general linear models (GLMs) to express the distance between a model and its sub model. This statistic allows one to detect locations where the smoothed point process residuals are reduced by adding the interesting covariates into the model. The second spatial statistic tries to capture local improvements in the shape of the predicted intensity caused by an interesting, continuous covariate. A simulation scheme resembling the permutation inference for GLMs is used to obtain the null distribution of the statistics. Thereafter, a Monte Carlo test with graphical interpretation (a global envelope test) is applied to the empirical and simulated statistic fields to determine the global significance of the covariate and the spatially significant areas. We study the empirical significance level and power of the test in different scenarios and, by applying the test to simulated and real point pattern data, show that the proposed statistics can be valuable for model construction.

通常通过Wald检验对参数点过程模型中的协变量效应进行检验，该检验依赖于检验统计量的渐近零分布。我们提出了测试的蒙特卡洛版本，该版本还允许对全局拟合模型中的协变量效应进行局部调查。为此，建议使用两种不同的测试统计量：第一种是在观察窗的每个位置计算的空间统计量，类似于经典的$F$-统计量，通常用于一般线性模型（GLM）中，以表示模型与其子模型之间的距离。通过统计，可以将有趣的协变量添加到模型中，从而检测出平滑点过程残差减少的位置。第二个空间统计量试图捕获由有趣的连续协变量引起的预测强度形状的局部改进。使用类似于GLM排列推理的模拟方案来获取统计信息的零分布。此后，将具有图形解释的蒙特卡洛检验（全局包络检验）应用于经验和模拟统计字段，以确定协变量和空间有效面积的全局重要性。