标题： 基于尺度经验K函数的点过程渐近拟合优度检验 显示英文标题

标题： Asymptotic Goodness-of-Fit Tests for Point Processes Based on Scaled Empirical K-Functions

摘要： 我们研究了尺度化边矫正的经验（广义）K-函数序列（修改了Ripley的K-函数），每个K-函数都是由一个在采样窗口\( W_n \)中的$d$-维四阶平稳点过程的单一观测构造的，且随着某种尺度速率无限增长（当\( n \to \infty \)时）。在一些自然假设下，证明了尺度化经验K-函数与尺度化理论K-函数之间的归一化差值弱收敛于均值为零的高斯过程，并具有简单的协方差函数。该结果表明了经验K-函数与理论K-函数之间的一种偏差度量，其极限分布已知，可用于基于点过程的强度和（广义）K-函数进行拟合优度检验。类似地，还推导出了测试统计量，用于检验\( W_n \)中的两个独立点过程是否具有相同分布，而无需明确知道它们的强度和K-函数。 显示英文摘要

摘要： We study sequences of scaled edge-corrected empirical (generalized) K-functions (modifying Ripley's K-function) each of them constructed from a single observation of a $d$-dimensional fourth-order stationary point process in a sampling window W_n which grows together with some scaling rate unboundedly as n --> infty. Under some natural assumptions it is shown that the normalized difference between scaled empirical and scaled theoretical K-function converges weakly to a mean zero Gaussian process with simple covariance function. This result suggests discrepancy measures between empirical and theoretical K-function with known limit distribution which allow to perform goodness-of-fit tests for checking a hypothesized point process based only on its intensity and (generalized) K-function. Similar test statistics are derived for testing the hypothesis that two independent point processes in W_n have the same distribution without explicit knowledge of their intensities and K-functions.