Point process models are a natural approach for modelling data that arise as point events. In the case of Poisson counts, these may be fitted easily as a weighted Poisson regression. Point processes lack the notion of sample size. This is problematic for model selection, because various classical criteria such as the Bayesian information criterion (BIC) are a function of the sample size, n, and are derived in an asymptotic framework where n tends to infinity. In this paper, we develop an asymptotic result for Poisson point process models in which the observed number of point events, m, plays the role that sample size does in the classical regression context. Following from this result, we derive a version of BIC for point process models, and when fitted via penalised likelihood, conditions for the LASSO penalty that ensure consistency in estimation and the oracle property. We discuss challenges extending these results to the wider class of Gibbs models, of which the Poisson point process model is a special case.

中文翻译：

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空间点过程的有效样本大小是多少？

点过程模型是对作为点事件出现的数据进行建模的自然方法。在泊松计数的情况下，这些可以很容易地拟合为加权泊松回归。点过程缺乏样本量的概念。这对于模型选择是有问题的，因为各种经典标准，例如贝叶斯信息标准 (BIC) 是样本大小n的函数，并且是在n趋于无穷大的渐近框架中导出的。在本文中，我们开发了泊松点过程模型的渐近结果，其中观察到的点事件数m, 扮演样本量在经典回归上下文中的作用。根据这个结果，我们为点过程模型推导出 BIC 版本，当通过惩罚似然拟合时，LASSO 惩罚的条件确保估计和预言机属性的一致性。我们讨论了将这些结果扩展到更广泛的 Gibbs 模型类别的挑战，其中 Poisson 点过程模型是一个特例。