Continuous determinantal point processes (DPPs) are a class of repulsive point processes on $\mathbb{R}^d$ with many statistical applications. Although an explicit expression of their density is known, this expression is too complicated to be used directly for maximum likelihood estimation. In the stationary case, an approximation using Fourier series has been suggested, but it is limited to rectangular observation windows and no theoretical results support it. In this contribution, we investigate a different way to approximate the likelihood by looking at its asymptotic behaviour when the observation window grows towards $\mathbb{R}^d$. This new approximation is not limited to rectangular windows, is faster to compute than the previous one, does not require any tuning parameter, and some theoretical justifications are provided. The performances of the associated estimator are assessed in a simulation study on standard parametric models on $\mathbb{R}^d$ and compare favourably to common alternative estimation methods for continuous DPPs.
翻译:连续的决定性点进程(DPPs) 是一种具有多种统计应用的 $\ mathbb{R ⁇ d$ 的令人厌恶的点进程类别。 虽然已知其密度的明显表达方式, 但这个表达方式过于复杂, 无法直接用于最大的可能性估计。 在固定的案例中, 已经建议使用 Fourier 序列近似值, 但是它仅限于矩形观测窗口, 没有理论结果支持它 。 在此贡献中, 我们用不同的方法来估计可能性, 以观察窗口在增长到$\ mathbb{R ⁇ d$时的无症状行为。 这个新的近似值并不局限于矩形窗口, 其计算速度比前一个窗口要快, 不需要任何调制参数, 并且提供了一些理论理由 。 相关的估计器的性能在对 $\mathb{R ⁇ d$ 的标准参数模型的模拟研究中得到评估, 并且与连续的 DPPP 通用的替代估算方法相比是有利的 。