We study Gaussian-copula models with discrete margins, with primary emphasis on low-count (Poisson) data. Our goal is exact yet computationally efficient maximum likelihood (ML) estimation in regimes where many observations contain small counts, which imperils both identifiability and numerical stability. We develop three novel Kendall's tau-based approaches for initialization tailored to discrete margins in the low-count regime and embed it within an inference functions for margins (IFM) inspired start. We present three practical initializers (exact, low-intensity approximation, and a transformation-based approach) that substantially reduce the number of ML iterations and improve convergence. For the ML stage, we use an unconstrained reparameterization of the model's parameters using the log and spherical-Cholesky and compute exact rectangle probabilities. Analytical score functions are supplied throughout to stabilize Newton-type optimization. A simulation study across dimensions, dependence levels, and intensity regimes shows that the proposed initialization combined with exact ML achieves lower root-mean-squared error, lower bias and faster computation times than the alternative procedures. The methodology provides a pragmatic path to retain the statistical guarantees of ML (consistency, asymptotic normality, efficiency under correct specification) while remaining tractable for moderate- to high-dimensional discrete data. We conclude with guidance on initializer choice and discuss extensions to alternative correlation structures and different margins.

翻译：我们研究了具有离散边缘分布的高斯Copula模型，主要关注低计数（泊松）数据。我们的目标是在大量观测值包含小计数的情形下实现精确且计算高效的最大似然（ML）估计，这种情形同时危及模型的可识别性和数值稳定性。我们针对低计数区间的离散边缘分布，开发了三种基于Kendall's tau的新型初始化方法，并将其嵌入受边缘推断函数（IFM）启发的起始框架中。我们提出了三种实用的初始化器（精确法、低强度近似法和基于变换的方法），显著减少了ML迭代次数并改善了收敛性。在ML阶段，我们采用对数与球面Cholesky变换对模型参数进行无约束重参数化，并计算精确的矩形概率。全程提供解析评分函数以稳定牛顿型优化。在不同维度、依赖水平和强度区间下的仿真研究表明，所提出的初始化方法结合精确ML估计，相比替代方案实现了更低的均方根误差、更小的偏差和更快的计算时间。该方法为保持ML的统计保证（一致性、渐近正态性、正确设定下的有效性）提供了一条实用路径，同时对于中高维离散数据保持可处理性。最后我们提供了初始化器选择的指导，并讨论了向替代相关结构和不同边缘分布的扩展。